text: White Swiss mouse, American Journal of Anatomy, 10.1002/aja.1000970105, 97, 1, (129-153), (2005). [33] Long. The adult male will weigh from 400-500 g with the female being about 100 g less. In 18th century Europe, wild brown rats ran rampant and this infestation fueled the industry of rat-catching. Other common strains are the Sprague Dawley, Fischer 344,[8] Holtzman albino strains, Long–Evans, and Lister black hooded rats. A scatterplot of the weight in grams and time since birth in weeks shows a fairly strong, positive linear relationship. Inbred strains of Rattus norvegicus used for scientific research, "Lab rat" redirects here. Humans can mimic these behaviors, and doing so helps lower the stress level of the rat. The Lewis rat was developed by Margaret Lewis from Wistar stock in the early 1950s. The shaking rat Kawasaki (SRK) is an autosomal recessive mutant rat that has a short deletion in the RELN (Reelin) gene. Specifications. (a) How many rads does the animal receive? Second, Lewis rats are prone to develop a spontaneous transplantable lymphatic leukaemia. The linear regression weight= 100+40(time) models the data fairly well. Many investigators who wish to trace observations on behavior and physiology to underlying genes regard aspects of these in rats as more relevant to humans and easier to observe than in mice, giving impetus to the development of genetic research techniques applicable to rat. Lastly, when in advanced age, they sometimes develop spontaneous glomerular sclerosis. Body Weight Information for C57BL/6J (000664) JAX ® Mice Strain - C57BL/6J. The spontaneously hypertensive rat and the Lewis rat are other well-known stocks developed from Wistar rats. [35] This results in the lowered expression of Reelin protein, essential for proper cortex lamination and cerebellum development. gestational age of lab animal embryos is vital to many areas of biomedical research. To change this deceivers, we need to know the R B E, which is given tow us as 0.75 So to find my Rams sequel to my rads times my R B E, which is 0.75 which is 1.10 Brems. The a verage weaning age for humans is. The red can describe how much energy or how many jewels of energy per kilogram, and one rad is equal 2.1 Jules her kilogram. Lab Supply has a number of specialized diets to make sure that your rat gets the exact nutrition you want. When working with young rats, placing a hand over the rat's back will cause it to jump into the palm of the handler allowing it to be grasped. Lesioning (destroying) which part of the brain in a laboratory rat will cause the rat to eat continuously and balloon to several times its normal weight? The first time one of these albino mutants was brought into a laboratory for a study was in 1828 for an experiment on fasting. By having this kind of population, it is possible to conduct experiments on the roles of genes, or conduct experiments that exclude variations in genetics as a factor. The word "strain" refers to a group of rats with a common ancestor. They are born without fur, their ears and eyes are both closed and weight only about 6 to 8 grams. Females tend to weigh around 250-450g while males can weigh between 400-800g. Data was collected on the weight of a male laboratory rat for the first 25 weeks after its birth. Rats are usually supplied feed free choice and they eat 10-30 g a day (5 g/100 g body weight/day). The Zucker rat was bred to be a genetic model for research on obesity and hypertension. Later in the 19th and 20th century they were bred for their coat color, and subsequently characterized genetically. WikiMatrix. (c) If the RBE is $0.75,$ what is the equivalent dose in $\mathrm{Sv} ?$, {'transcript': "to measure exposure to radiation. While less commonly used for research than mice, rats have served as an important animal model for research in psychology and biomedical science.[1]. And one rim is equal 2.1 Sieverts. This page was last edited on 2 December 2020, at 15:14. The linear regression equation is y= 40x+100, where x is number of weeks and y is weight in grams. [22] Thus, 180 ÷ 21 = 8.6 human da ys = 1 rat day and. So my energy per kilograms is 3.894 times 10 to the minus three jewels divided by 0.265 kilograms, which equals 1.47 times. Water is supplied free choice and they usually drink 20-50 ml a day (10 ml/100 g body … Puppy Weight Calculator Interactive Tool. What does the y-intercept mean in context of the problem? Play this game to review Algebra I. The mouse is probably the most genetically and biologically characterized mammal in the world. All laboratory rats are part of the same species: the Norway, or brown, rat. More than 100 million mice and rats are killed in U.S. laboratories every year. The animal that's being exposed to this has a mass of 200 and 65 grams, which converted two kilograms by dividing by thousands is 0.265 kilograms. The Laboratory Rat, Second Edition features updated information on a variety of topics including: rat genetics and genomics, both spontaneous and induced disease; state-of-the-art technology for housing and husbandry; occupational health, and experimental models. And they aren’t just big mice. Click 'Join' if it's correct. It has a naturally occurring genetic mutation that makes specimens unable to produce the hormone vasopressin, which helps control kidney function. the Laboratory Rat.1 He was known to say, “Rats are just little people without socks and shoes.” His research and that of others confirmed much of what owners of pet rats have long known to be true: Rats are intelligent, social animals. You can also offer rawhide chew sticks or hard dog biscuits. It is important that animals are habituated to this method of handling from an early age. Much of the genome of Rattus norvegicus has been sequenced. [6] The genetics of rats was studied by William Ernest Castle at the Bussey Institute of Harvard University until it closed in 1994. [3], The rat found early use in laboratory research in five areas: W. S. Small suggested that the rate of learning could be measured by rats in a maze; a suggestion employed by John B. Watson for his Ph.D. dissertation in 1903. It is actually quite an easy piece of equipment to use, although one needs to be careful when reading the grams off the dial. Rats are very social and physical interactions are part of socialization for them, especially when they are young (if there are group-housed rats take a moment to watch them on the rack - lots of tumbling/tackling in adolescent groups, and older rats groom and climb over each other). Wistar rats are more active than others like Sprague Dawley rats. There are two types of Zucker rat: a lean Zucker rat, denoted as the dominant trait (Fa/Fa) or (Fa/fa); and the characteristically obese (or fatty) Zucker rat, which is actually a recessive trait (fa/fa) of the leptin receptor, capable of weighing up to 1 kilogram (2.2 lb)—more than twice the average weight. The Wistar rat is currently one of the most popular rats used for laboratory research. By contrast, outbred populations are used when identical genotypes are unnecessary or a population with genetic variation is required, and these rats are usually referred to as stocks rather than strains. Adult rats are gently grasped around the thorax, lifted, and quickly placed into another cage. In about one week, they can put on 10 more grams and look completely different, including showing some subtle fur and pigment to it! In rats, this is accomplished through inbreeding. When you put the lid back on the basket be careful not to catch the rat's … The Sprague Dawley rat is an outbred multipurpose breed of albino rat used extensively in medical and nutritional research. Rats are bred specifically for laboratory and testing purposes, and certain strains are more desirable for different experiments. A laboratory rat weighs 265 \mathrm{~g} and absorbs 1.77 \times 10^{10} \beta^{-} particles, each with an energy of 2.20 \times 10^{-13} \mathrm{~J}. "[10], During food rationing due to World War II, British biologists ate laboratory rat, creamed. Although the genetic defect was not known for many years, it was identified in the year 2000 as a mutation in the gene MERTK. Wiley Online Library . The most common are adenomas of the pituitary and adenomas/adenocarcinomas of the adrenal cortex in both sexes, mammary gland tumors and endometrial carcinomas in females, and C-cell adenomas/adenocarcinomas of the thyroid gland and tumors of the haemopoietic system in males. Tend to the minus two Jules per kilogram. The name was originally hyphenated, although the brand styling today (Sprague Dawley, the trademark used by Envigo) is not. Specifications › Contact a Product Specialist › Animal Weighing Scale. "}, EMAILWhoops, there might be a typo in your email. WikiMatrix. Some data were collected on the weight of a male white laboratory rat for the first 25 weeks after its birth. Fuzzy rats were identified in 1976 in a Pennsylvania lab. [3] As there is evidence that the hooded rat was known as the "Japanese rat" in the early 20th century, Kuramoto concluded that one or more Japanese hooded rats might have been brought to Europe or the Americas and an albino rat that emerged as a product of the breeding of these hooded rats was the common ancestor of all the albino laboratory rats in use today. (a) How m… Scientists have bred many strains or "lines" of rats specifically for experimentation. By transmitting high-energy waves we think it may be possible to deafen laboratory rats. [9] In October 2003, researchers succeeded in cloning two laboratory rats by nuclear transfer. Size/weight will vary markedly between strains. COVID-19 Customer Information; ARC Online Order Payments; Online Ordering. So a mouse of 20g = 1.2-1.6mL. Inbred strains are also available, but are not as commonly used as inbred mice. the biobreeding diabetes-prone rat or BBDP rat) is an inbred strain that spontaneously develops autoimmune type 1 diabetes. [33], Current research applications include transplantation research, induced arthritis and inflammation, experimental allergic encephalitis, and STZ-induced diabetes. Another unit that we use is the gray. The Sprague Dawley rat and Long–Evans rat were developed from Wistar rats. The historical importance of this species to scientific research is reflected by the amount of literature on it: roughly 50% more than that on laboratory mice. Note: you will not be able to collect this much blood as this is the total blood volume. [4] The first rat colony in America used for nutrition research was started in January 1908 by Elmer McCollum[5] and then, nutritive requirements of rats were used by Thomas Burr Osborne and Lafayette Mendel to determine the details of protein nutrition. Male and female mice were weighed the same day each week in nine mouse rooms representing all three breeding facilities. All three rats follow a similar pattern of weight gain, with very rapid gain up to three and a half months of age followed by a slower increase. So we divide 1.47 times 10 to the minus two by 20.1 We get 1.47 rad's to change this to graze, we just use the relationship between rads and graze or one rad is equal 2.1 graze or 1.47 times 10 to the minus two graze. Oh no! For other uses, see, "Use of House Mice in Biomedical Research", "43rd Annual Pathology of Laboratory Animals Course", Comparison of Neoplasms in Six Sources of Rats, "Rules and Guidelines for Nomenclature of Mouse and Rat Strains", "The Wistar Institute Archives: Rats (Not Mice) and History", "Studies on the effect of experimental nonketotic diabetes mellitus on antibacterial defense. They are utilized as a multipurpose model organism, frequently in behavioral and obesity research. The rats were being raised for laboratory use by Dr. Henry Schroeder and technician Tim Vinton, who noticed that the litter of 17 drank and urinated excessively. The first thing to do is remove the cover from the metal basket. This was the first in a series of developments that have begun to make rats tractable as genetic research subjects, although they still lag behind mice, which lend themselves better to the embryonic stem cell techniques typically used for genetic manipulation. A knockout rat (also spelled knock out or knock-out) is a genetically engineered rat with a single gene turned off through a targeted mutation. Long-Evans rats are white with a black hood, or occasionally white with a brown hood. Its phenotype is similar to the widely researched reeler mouse. But more on that later. Over time, breeding the rats for these contests may have produced variations in color, notably the albino and hooded varieties. [40] Obesity in Zucker rats is primarily linked to their hyperphagic nature and excessive hunger; however, food intake does not fully explain the hyperlipidemia or overall body composition.[38][40]. Provide tunnels, igloos, ropes, ladders, blocks, and child or baby safe toys for your rat to play with. All but one of the testicular tumors occurred in the rats from a single supplier. The pelleted feed is supplied as regular, breeder, certified, irradiated or autoclavable. [36] This and the Lewis rat are well-known stocks developed from Wistar rats. [37][38][39], Obese Zucker rats have high levels of lipids and cholesterol in their bloodstream, are resistant to insulin without being hyperglycemic, and gain weight from an increase in both the size and number of fat cells. Mice were fed a diet containing … They grow very fast though! The production of knockout rats became technically feasible in 2008, through work financed by $120 million in funding from the National Institutes of Health (NIH) via the Rat Genome Sequencing Project Consortium, and work accomplished by the members of the Knock Out Rat Consortium (KORC). Average Baby Rat Weight. PLace a large plastic bowl/ beaker on the weighing scales, tare the scales back to 0, take the bowl off, place your animal gently in the bowl and place it back on the scales. 3 weeks for laboratory rats (~P21). The strain re-capitulates many of the features of human type 1 diabetes and has contributed greatly to the research of T1DM pathogenesis.[28]. They are abused in everything from toxicology tests (in which they are slowly poisoned to death) to painful burn experiments to psychological experiments that induce terror, anxiety, depression, and helplessness. A laboratory rat or lab rat is a brown rat of the subspecies Rattus norvegicus domestica which is bred and kept for scientific research. Wistar rats (220–250 g body weight) were purchased from the laboratory animal center of Heilongjiang University of Chinese Medicine (Harbin, China) and divided into two groups (group A, drug group for dosed rat plasma, n = 7, and group B, control group for blank rat plasma, n = 7) and kept in a breeding room. Lithium battery included. It is characterized by its wide head, long ears, and a tail length that is always less than its body length. The Jackson Laboratory; Strain Summary; Disclaimer; Ordering. A 1972 study compared neoplasms in Sprague Dawley rats from six different commercial suppliers and found highly significant differences in the incidences of endocrine and mammary tumors. Quantity. [29] The more common ones are denoted as rnu (Rowett nude), fz (fuzzy), and shn (shorn). The leading cause of death among fz/fz rats is ultimately a progressive kidney failure that begins around the age of 1 year. They’re roughly the size of a coin even! Long and Evans in 1915 by crossing several Wistar females with a wild gray male. There were even significant variations in the incidences of adrenal medulla tumors among rats from the same source raised in different laboratories. Our ram is equal to the rad times a constant called the are B E. And finally we can measure radiation exposure by relating rams to a final unit called the Sea Vert. Click 'Join' if it's correct, By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy, Whoops, there might be a typo in your email. After the third month their weight gain continued but at a slower rate. Snip is consistently the lightest rat. Then, lift your rat from his homecage and deposite him in the basket. Over the next 30 years, rats were used for several more experiments and eventually the laboratory rat became the first animal domesticated for purely scientific reasons. The rats gained weight rapidly up to their third month. Characteristics include albino coloring, docile behavior, and low fertility. A laboratory rat or lab rat is a brown rat of the subspecies Rattus norvegicus domestica which is bred and kept for scientific research. Online Ordering Instructions; General Terms and Conditions for the Sale of Animals. Similar behavior has been observed in laboratory rats in experiments looking at spinal cord and peripheral nerve injuries. The researchers carried out experiments by observing laboratory rats as they wandered through a small open compartment. What does the y-intercept mean in context of the problem? So the total energy from these particles is 2.2 times 10 to the minus 13 times 1.77 times 10 to the 10 or three point 89 for times 10 to the minus three jewels. While less commonly used for research than mice, rats have served as an important animal model for research in psychology and biomedical science. However, since these rats are known to grow tumors at a high (and very variable) rate, the study was considered flawed in design and its findings unsubstantiated. Here are the actual weight measurements of the three rats. The laboratory rat, like its wild counterpart, is an extremely intelligent animal, probably more so than the other rodents commonly used for biomedical research. The reproductive function of rats was studied at the Institute for Experimental Biology at the University of California, Berkeley by Herbert McLean Evans and Joseph A. Data was collected on the weight of a male laboratory rat for the first 25 weeks after its birth. This mutation results in defective retinal pigment epithelium phagocytosis of photoreceptor outer segments.[34]. This breed was developed at the Wistar Institute in 1906 for use in biological and medical research, and is notably the first rat developed to serve as a model organism at a time when laboratories primarily used the house mouse (Mus musculus). The average blood volume for a laboratory mouse is ~6-8% of total body weight. Answer to A laboratory rat weighs 265 g and absorbs 1.77×1010 β− particles, each with an energy of 2.20×10−13 J. Mice were first used for the study of respiration in the 17th century. Domestic rats differ from wild rats in many ways: they are calmer and significantly less likely to bite, they can tolerate greater crowding, they breed earlier and produce more offspring, and their brains, livers, kidneys, adrenal glands, and hearts are smaller. There are two types of Zucker rat: a lean Zucker rat, denoted as the dominant trait (Fa/Fa) or (Fa/fa); and the characteristically obese (or fatty) Zucker rat, which is actually a recessive trait (fa/fa), capable of weighing up to 1 kilogram (Template:Convert/LoffAonSoff)Template:Convert/test/A—more than twice the average … If you have a mix-breed dog, see our puppy weight charts to see which breed category your puppy falls into based on his current weight and age. Stock Number 000664. We can relate the gray to the rad because one rad is equal to point 01 craze to symbolize G y. So our energy of each particle is 2.2 zero times 10 to the minus 13 jewels for particle, and the exposure includes 1.77 times 10 to the 10 beta particles. approximately 6 months (180 da ys), [21] while it is. Knockout rat disease models for Parkinson's disease, Alzheimer's disease, hypertension, and diabetes, using zinc-finger nuclease technology, are being commercialized by SAGE Labs. Shorn rats were bred from Sprague Dawley rats in Connecticut in 1998. Rats have long been used in cancer research; for instance at the Crocker Institute for Cancer Research.[7]. The average litter size of the Sprague Dawley rat is 11.0. David J. Sullivan, The Effect of Exsanguination on Organ Weight of Rats, Toxicologic Pathology, 10.1177/019262338501300310, 13, 3, ... Body‐weight ratios for certain organs of laboratory animals. Whether you are looking for irradiated diets, certified diets, or simple standard diets for your lab rats, we have exactly what you are looking for. The linear regression equation is y= 40x+100, where x is number of weeks and y is weight in grams. The authors of the study "stressed the need for extreme caution in evaluation of carcinogenicity studies conducted at different laboratories and/or on rats from different sources. Like NOD mice, biobreeding rats are used as an animal model for Type 1 diabetes. Hairless laboratory rats provide researchers with valuable data regarding compromised immune systems and genetic kidney diseases. 0 0 1. If you’re trying to help your rat lose weight, making sure it has plenty of activities will help it stay healthy and happy. . [2] Laboratory rats are frequently subject to dissection or microdialysis to study internal effects on organs and the brain, such as for cancer or pharmacological research. (And no, that doesn't mean they're all brown.) In the meantime, our AI Tutor recommends this similar expert step-by-step video covering the same topics. OpenSubtitles2018.v3. So in this particular example, we have a certain amount of energy and a certain number of particles, um, and a specific mass of an animal that's being exposed so we can start by finding the energy per kilogram. And then we change that to see Virts, knowing that one room is equal 2.1 seabirds. More than half of all laboratory rat strains are descended from the original colony established by physiologist Henry Donaldson, scientific administrator Milton J. Greenman, and genetic researcher/embryologist Helen Dean King.[19][20]. One is called the Rad. Most are derived from the albino Wistar rat, which is still widely used. (b) What is this dose in Gy? [2], In Japan, there was a widespread practice of keeping rats as a domesticated pet during the Edo period and in the 18th century guidebooks on keeping domestic rats were published by Youso Tamanokakehashi (1775) and Chingan Sodategusa (1787). Badger & Moose getting to do some of the things they hadn't experienced before they came to live with me. [11][12][13][14][15][16], A strain, in reference to rodents, is a group in which all members are, as nearly as possible, genetically identical. Salter Stainless Steel Digital Animal Weighing Scale (3 kg Capacity) SCL-1015$75. The laboratory rat is a domesticated Norway rat, which in nature is one of the most widespread and abundant of the more than 70 species of the genus Rattus (family Muridae). Feed may be pelleted or powdered. Amendments to Orders (COVID-19) General Ordering Information. Adult rats can weigh anywhere from 250 - 500 grams, with males being larger than females. The researchers found that the incidence of tumors in Sprague-Dawley rats from different commercial sources varied as much from each other as from the other strains of rats.


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Functional Logic • Inquiry and Analogy • 9

Inquiry and Analogy • Dewey’s “Sign of Rain” • An Example of Inquiry

We turn again to Dewey’s vignette, tracing figures of logic on grounds of semiotic.

A man is walking on a warm day.  The sky was clear the last time he observed it;  but presently he notes, while occupied primarily with other things, that the air is cooler.  It occurs to him that it is probably going to rain;  looking up, he sees a dark cloud between him and the sun, and he then quickens his steps.  What, if anything, in such a situation can be called thought?  Neither the act of walking nor the noting of the cold is a thought.  Walking is one direction of activity;  looking and noting are other modes of activity.  The likelihood that it will rain is, however, something suggested.  The pedestrian feels the cold;  he thinks of clouds and a coming shower.

(John Dewey, How We Think, 6–7)

Inquiry and Inference

If we follow Dewey’s “Sign of Rain” example far enough to consider the import of thought for action, we realize the subsequent conduct of the interpreter, progressing up through the natural conclusion of the episode — the quickening steps, seeking shelter in time to escape the rain — all those acts form a series of further interpretants, contingent on the active causes of the individual, for the originally recognized signs of rain and the first impressions of the actual case.  Just as critical reflection develops the associated and alternative signs which gather about an idea, pragmatic interpretation explores the consequential and contrasting actions which give effective and testable meaning to a person’s belief in it.

Figure 10 charts the progress of inquiry in Dewey’s “Sign of Rain” example according to the stages of reasoning identified by Peirce, focusing on the compound or mixed form of inference formed by the first two steps.

$\text{Figure 10. Cycle of Inquiry}$

• Step 1 is Abductive, abstracting a Case from the consideration of a Fact and a Rule.
• $\begin{array}{lll} \textsc{Fact} & : & {C \Rightarrow A}, \end{array}$     In the Current situation the Air is cool.
• $\begin{array}{lll} \textsc{Rule} & : & {B \Rightarrow A}, \end{array}$     Just Before it rains, the Air is cool.
• $\begin{array}{lll} \textsc{Case} & : & {C \Rightarrow B}, \end{array}$     The Current situation is just Before it rains.
• Step 2 is Deductive, admitting the Case to another Rule and arriving at a novel Fact.
• $\begin{array}{lll} \textsc{Case} & : & {C \Rightarrow B}, \end{array}$     The Current situation is just Before it rains.
• $\begin{array}{lll} \textsc{Rule} & : & {B \Rightarrow D}, \end{array}$     Just Before it rains, a Dark cloud will appear.
• $\begin{array}{lll} \textsc{Fact} & : & {C \Rightarrow D}, \end{array}$     In the Current situation, a Dark cloud will appear.

What precedes is nowhere near a complete analysis of Dewey’s example, even so far as it might be carried out within the constraints of the syllogistic framework, and it covers only the first two steps of the inquiry process, but perhaps it will do for a start.

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# Differentiation of function

There is a passage in a physics textbook I don't quite follow. Since my question is mathematical, I've decided to post it here. The book says:

Let $V$ be the volume of a molecule and assume $V = nr^3$. Then, because the incompressibility $K$ is defined as $K = -V \frac{\partial P}{\partial V}$, and pressure $P$ is defined as $P = - \frac{\partial U}{\partial V}$, one finds $K =V \frac{\partial ^2 U}{\partial V^2}$.

Using $V = nr^3$, $P = -\frac{dU}{dr} \frac{dr}{dV} = - \frac{1}{3nr^2} \frac{dU}{dr}$. Thus the incompressibility becomes

$$K = -nr^3 \frac{dP}{dr} \frac{dr}{dV} = \frac{r}{9n} \frac{d}{dr} \left[\frac{1}{r^2} \frac{dU}{dr} \right]$$

Question 1

I don't quite see how this last expression is obtained. I know that since $V = nr^3$ we have $\frac{dV}{dr} = 3nr^2$, so $\frac{dr}{dV} = \frac{1}{3nr^2}$. Thus, I see how the expression for $P$ is obtained. Also I see that for $K$ we have $K = -nr^3 \frac{d}{dr} \left[ -\frac{1}{3nr^2} \frac{dU}{dr} \right] \frac{dr}{dV}$, but I have a hard time working this out algebraically to get the expression above. If anyone can show me the intermediate steps here, then I would be very grateful.

The book further states:

At the equilibrium position $r = r_0$, $\frac{dU}{dr} = 0$. Thus

$$K_0 = \frac{1}{9nr_0} \left(\frac{d^2 U}{dr^2} \right)_0$$

Question 2

This confuses me. If $\frac{dU}{dr} = 0$ and $K = \frac{r}{9n} \frac{d}{dr} \left[\frac{1}{r^2} \frac{dU}{dr} \right]$, the shouldn't we get simply $K = 0$?

Any help on any of these questions will really be appreciated!

1. $$K=-V\frac{\partial P}{\partial V}=-V\frac{\partial P}{\partial r}\frac{\partial r}{\partial V} =-nr^3\frac{\partial P}{\partial r}\frac{\partial r}{\partial V} =-nr^3\frac{\partial P}{\partial r}\frac{1}{3nr^2}\\ =-nr^3\frac{1}{3nr^2}\frac{\partial}{\partial r}\left[-\frac1{3nr^2}\frac{\partial U}{\partial r}\right] =\frac{r}{9n}\frac{\partial}{\partial r}\left[\frac1{r^2}\frac{\partial U}{\partial r}\right]$$ 2. $$\frac{\partial}{\partial r}\left[\frac1{r^2}\frac{\partial U}{\partial r}\right]=-\frac2{r^3}\frac{\partial U}{\partial r}+\frac1{r^2}\frac{\partial^2 U}{\partial r^2}$$

• Great! Thanks a lot. Really appreciate it! – Kristian Jul 1 '14 at 18:22

\begin{align} K&= V\,\partiald[2]{U}{V}\ =\ \overbrace{nr^{3}}^{\ds{=\ V\quad}}\ \overbrace{% {1 \over 9n^{2}r^{2}}\,\partiald{}{r}\pars{{1 \over r^{2}}\,\partiald{U}{r}}} ^{\ds{=\ \partiald[2]{U}{V}}}\ =\ {r \over 9n}\,\,\partiald{}{r}\pars{{1 \over r^{2}}\,\partiald{U}{r}} \\[3mm]&={r \over 9n}\braces{% \partiald{\pars{1/r^{2}}}{r}\,\partiald{U}{r} +{1 \over r^{2}}\,\partiald{}{r}\bracks{\partiald{U}{r}}} =-\,{2 \over 9nr^{2}}\,\partiald{U}{r} + {1 \over 9nr}\,\partiald[2]{U}{r} \end{align}

$\ds{\large\bf\mbox{Question 2}}$:

$\ds{\pars{\partiald[2]{U}{r}}_{0}}$ is a 'short notation': The second derivative is evaluated at the point where the first derivative is zero. It doesn't need to be zero. For example: $$\fermi\pars{x}\equiv x^{2} - 2x\,,\qquad \fermi'\pars{1} = 0\,,\qquad\fermi''\pars{1} = 2 \not= 0$$

• Thanks a lot! Much appreciated! Wish I could mark both answers as accepted! – Kristian Jul 1 '14 at 18:23


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## Intermediate Algebra: Connecting Concepts through Application

$\text{LCD: } 84x^2y \\\\\text{Equivalent of 1st Expression: } \dfrac{49}{84x^2y} \\\\\text{Equivalent of 2nd Expression: } \dfrac{9x}{84x^2y}$
$\bf{\text{Solution Outline:}}$ To find the $LCD$ of the given expressions, $\dfrac{7}{12x^2y}$ and $\dfrac{3}{28xy} ,$ express the denominators in factored form. Then take the highest exponent of each factor. Finally use the $LCD$ to find the required equivalent expression for the given. $\bf{\text{Solution Details:}}$ In factored form, the given expressions are equivalent to \begin{array}{l}\require{cancel} \dfrac{7}{2^2(3)(x^2)(y)} \text{ and } \dfrac{3}{2^2(7)(x)(y)} .\end{array} Taking the highest exponent of each factor, then the $LCD$ is \begin{array}{l}\require{cancel} 2^2(3)(7)(x^2)(y) \\\\= 84x^2y .\end{array} Multiplying the given expression by an expression equal to $1$ such that the denominator becomes the $LCD,$ then the given expressions are equivalent to \begin{array}{l}\require{cancel} \dfrac{7}{12x^2y}\cdot\dfrac{7}{7}= \dfrac{49}{84x^2y} \\\\\text{ and }\\\\ \dfrac{3}{28xy}\cdot\dfrac{3x}{3x}= \dfrac{9x}{84x^2y} .\end{array} Hence, the needed pieces of information are as follows: \begin{array}{l}\require{cancel} \text{LCD: } 84x^2y \\\\\text{Equivalent of 1st Expression: } \dfrac{49}{84x^2y} \\\\\text{Equivalent of 2nd Expression: } \dfrac{9x}{84x^2y} .\end{array}


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### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

# Collatz-ish

##### Stage: 3 and 4 Short Challenge Level:

The terms of the sequence are

$$6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, \dots$$

As can be seen, there will now be no other terms in the sequence other than $4$, $2$, and $1$. It can also be seen that the only values of $n$ for which the $n$th term equals $n$ are $13$ and $16$.

This problem is taken from the UKMT Mathematical Challenges.
View the archive of all weekly problems grouped by curriculum topic

View the previous week's solution
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# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark 2.1.4.23 (Adjoint Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be nonunital monoidal categories and suppose we are given a pair of adjoint functors $\xymatrix@1{\operatorname{\mathcal{C}} \ar@ <.4ex>[r]^-{F} & \operatorname{\mathcal{D}} \ar@ <.4ex>[l]^-{G}}$, so that we have an isomorphism of oriented fiber products $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ (see Notation 2.1.4.19). Applying Proposition 2.1.4.21 (and the dual characterization of nonunital colax monoidal functors), we see that the following are equivalent:

• The datum of a nonunital lax monoidal structure on the functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.

• The datum of a nonunital colax monoidal structure on the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.

• The datum of a nonunital monoidal structure on the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\simeq \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}$ which is compatible with the nonunital monoidal structures on $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ (meaning that the projection map $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\times \operatorname{\mathcal{D}}$ is a nonunital strict monoidal functor).


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Time Limit : sec, Memory Limit : KB
English

## Problem A Secret of Chocolate Poles

Wendy, the master of a chocolate shop, is thinking of displaying poles of chocolate disks in the showcase. She can use three kinds of chocolate disks: white thin disks, dark thin disks, and dark thick disks. The thin disks are $1$ cm thick, and the thick disks are $k$ cm thick. Disks will be piled in glass cylinders.

Each pole should satisfy the following conditions for her secret mission, which we cannot tell.

• A pole should consist of at least one disk.
• The total thickness of disks in a pole should be less than or equal to $l$ cm.
• The top disk and the bottom disk of a pole should be dark.
• A disk directly upon a white disk should be dark and vice versa.

As examples, six side views of poles are drawn in Figure A.1. These are the only possible side views she can make when $l = 5$ and $k = 3$.

Figure A.1. Six chocolate poles corresponding to Sample Input 1

Your task is to count the number of distinct side views she can make for given $l$ and $k$ to help her accomplish her secret mission.

### Input

The input consists of a single test case in the following format.

$l$ $k$

Here, the maximum possible total thickness of disks in a pole is $l$ cm, and the thickness of the thick disks is $k$ cm. $l$ and $k$ are integers satisfying $1 \leq l \leq 100$ and $2 \leq k \leq 10$.

### Output

Output the number of possible distinct patterns.

### Sample Input 1

5 3

### Sample Output 1

6

### Sample Input 2

9 10

### Sample Output 2

5

### Sample Input 3

10 10

### Sample Output 3

6

### Sample Input 4

20 5

### Sample Output 4

86

### Sample Input 5

100 2

### Sample Output 5

3626169232670


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Math Help - Find derivative using Part 1 of the Fundamental Theorem of Calculus?

1. Find derivative using Part 1 of the Fundamental Theorem of Calculus?

I never seen this problem before! I tried expanding and doing all that stuff but I got the wrong answer!

The function is (3+v^2)^10 with limits of integration [sin(x), cos(x)]

2. Fundamental theorem of calculus - Wikipedia, the free encyclopedia

Just in case a picture helps...

... differentiating downwards with respect to x (on the left) or with respect to the dashed balloon (right), the latter referring to chain rule, as below.

You might like to use properties of definite integrals to split the integral around a constant a.

For the half from a up to cos x...

... where (key in spoiler) ...

Spoiler:

... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case ), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

So you have half of the derivative you seek, on the bottom row, and you'll want to subtract a similar result got from applying the same process but with sin in the dashed balloon.
_________________________________________

Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!


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5
Q:

# A train covers a distance in 50 minutes, if it runs at a speed of 48kmph on an average. Find the speed at which the train must run to reduce the time of journey to 40 minutes.

 A) 50 km/hr B) 60 km/hr C) 65 km/hr D) 70 km/hr

Explanation:

We are having time and speed given, so first we will calculate the distance. Then we can get new speed for given time and distance.
Lets solve it.

Time = 50/60 hr = 5/6 hr
Speed = 48 mph
Distance = S*T = 48 * 5/6 = 40 km

New time will be 40 minutes so,
Time = 40/60 hr = 2/3 hr
Now we know,
Speed = Distance/Time
New speed = 40*3/2 kmph = 60kmph

Q:

A person can row 8 1/2 km an hour in still water and he finds that it takes him twice as long to row up as to row down the river. The speed of the stream ?

 A) 1.78 kmph B) 2.35 kmph C) 2.83 kmph D) 3.15 kmph

Explanation:

Given speed of the person = 8 1/2 = 17/2 kmph
Let the speed of the stream = x kmph
speed of upstream = 17/2 - x
speed of downstream = 17/2 + x
But given that,
2(17/2 - x) = 17/2 + x
=> 3x = 17/2
=> x = 2.83 kmph.

4 51
Q:

Karthik could cover a distance of 200 km in 22 days while resting for 2 hrs. per day. In how many days (approx) he will cover a distance of 250 km while resting for 2 hrs. per day and moving with 2/3rd of the previous speed ?

 A) 41.25 days B) 37.5 days C) 39.75 days D) 40 days

Explanation:

Given Karthik can cover the distance of 200 kms resting 2 hrs per day in 22 days.

Let the initial speed be '1'

Hence, Time = 22(24hrs - 2)

Noe new speed = 2/3

Let the number of days he take be 'D'

Therefore, $\inline \fn_jvn \small \frac{22(24-2)x3}{200}=\frac{D(24-2)x2}{250}$

D = 11 x 3 x 5/4 = 41.25 days. = 41 1/4 days.

7 102
Q:

A bus covers its journey at the speed of 80km/hr in 10hours. If the same distance is to be covered in 4 hours, by how much the speed of bus will have to increase ?

 A) 85 km/hr B) 95 km/hr C) 105 km/hr D) 120 km/hr

Explanation:

Initial speed = 80km/hr
Total distance = 80 x 10 = 800km
New speed = 800/4 =200km/hr
Increase in speed = 200 - 80 = 120km/hr

6 168
Q:

Joel travels the first 3 hours of his journey at 60 mph speed and the remaining 5 hours at 24 mph speed. What is the average speed of Joel's travel in kmph ?

 A) 38 kmph B) 40 kmph C) 60 kmph D) 54 kmph

Explanation:

Average speed = Total distance / Total time.

Total distance traveled by Joel = Distance covered in the first 3 hours + Distance covered in the next 5 hours.

Distance covered in the first 3 hours = 3 x 60 = 180 miles
Distance covered in the next 5 hours = 5 x 24 = 120 miles

Therefore, total distance traveled = 180 + 120 = 300 miles.

Total time taken = 3 + 5 = 8 hours.

Average speed = 300/8 = 37.5 mph.

we know that 1 mile = 1.6 kms

=> 37.5 miles = 37.5 x 1.6 = 60 kms

Average speed in Kmph = 60 kmph.

2 155
Q:

Two persons start running simultaneously around a circular track of length 600 m from the same point at speeds of 25 kmph and 35 kmph. When will they meet for the first time any where on the track  ?

 A) 54 sec B) 36 sec C) 72 sec D) 11 sec

Explanation:

Time taken to meet the first time = length of track/relative speed

Given the length of the track is 600 m

The relative speed = 25 + 35 = 60 kmph = 60 x 5/18 m

Therefore Time = $\inline \fn_jvn \frac{600}{(25+35)\frac{5}{18}}$  = 600/60 x (18/5) = 36 sec.


text: Let us take an example to understand the concept, For an equilateral triangle, n = 3. $$(\red 6 -2) \cdot 180^{\circ} = (4) \cdot 180^{\circ}= 720 ^{\circ}$$. nt. A regular polygon is simply a polygon whose sides all have the same length and, (a polygon with sides of equal length and angles of equal measure), Finding 1 interior angle of a regular Polygon, $$\angle A \text{ and } and \angle B$$. Use Interior Angle Theorem:$$(\red 5 -2) \cdot 180^{\circ} = (3) \cdot 180^{\circ}= 540 ^{\circ}$$. Exterior angle of regular polygon is given by \frac { { 360 }^{ 0 } }{ n } , where “n” is number of sides of a regular polygon. Formula to find 1 angle of a regular convex polygon of n sides =, $$\angle1 + \angle2 + \angle3 + \angle4 = 360°$$, $$\angle1 + \angle2 + \angle3 + \angle4 + \angle5 = 360°$$. An exterior angle on a polygon is formed by extending one of the sides of the polygon outside of the polygon, thus creating an angle supplementary to the interior angle at that vertex. BE / CE = AB / AC. If each exterior angle measures 10°, how many sides does this polygon have? Following the formula we have: 360 degrees / 6 = 60 degrees. The measure of each interior angle of an equiangular n -gon is. To make the process less tedious, the sum of interior angles in all regular polygons is calculated using the formula given below: Sum of interior angles = (n-2) x 180°, here n = here n = total number of sides. A quadrilateral has 4 sides. Substitute 12 (a dodecagon has 12 sides) into the formula to find a single exterior angle. An interior angle would most easily be defined as any angle inside the boundary of a polygon. By using this formula, easily we can find the exterior angle of regular polygon. 3) The measure of an exterior angle of a regular polygon is 2x, and the measure of an interior angle is 4x. The sides of the angle are those two rays. Angle and angle must each equal degrees. so, angle ADC = (180-x) degrees. 1) In the given figure, AE is the bisector of the exterior ∠CAD meeting BC produced in E. If AB = 10 cm, AC = 6 cm and BC = 12 cm, find CE. The opposite interior angles must be equivalent, and the adjacent angles have a sum of degrees. Polygons come in many shapes and sizes. Regular Polygon : A regular polygon has sides of equal length, and all its interior and exterior angles are of same measure. An exterior angle of a polygon is made by extending only one of its sides, in the outward direction. Substitute 10 (a decagon has 10 sides) into the formula to find a single exterior angle. Because of the congruence of vertical angles, it doesn't matter which side is extended; the exterior angle will be the same. You can also use Interior Angle Theorem:$$(\red 3 -2) \cdot 180^{\circ} = (1) \cdot 180^{\circ}= 180 ^{\circ}$$. Calculate the measure of 1 interior angle of a regular hexadecagon (16 sided polygon)? Measure of a Single Exterior Angle Angles: re also alternate interior angles. An exterior angle of a triangle is formed by any side of a triangle and the extension of its adjacent side. Six is the number of sides that the polygon has. Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. $(n-2)\cdot180^{\circ}$. Substitute 12 (a dodecagon has 12 sides) into the formula to find a single interior angle. Notice that corresponding interior and exterior angles are supplementary (add to 180°). Regardless, there is a formula for calculating the sum of all of its interior angles. And since there are 8 exterior angles, we multiply 45 degrees * 8 and we get 360 degrees. If you learn the formula, with the help of formula we can find sum of interior angles of any given polygon. It's possible to figure out how many sides a polygon has based on how many degrees are in its exterior or interior angles. exterior angles. For a triangle: The exterior angle dequals the angles a plus b. Know the formula from which we can find the sum of interior angles of a polygon.I think we all of us know the sum of interior angles of polygons like triangle and quadrilateral.What about remaining different types of polygons, how to know or how to find the sum of interior angles.. Substitute 8 (an octagon has 8 sides) into the formula to find a single interior angle. Let, the exterior angle, angle CDE = x. and, it’s opposite interior angle is angle ABC. Exterior Angles The diagrams below show that the sum of the measures of the exterior angles of the convex polygon is 360 8. The moral of this story- While you can use our formula to find the sum of the interior angles of any polygon (regular or not), you can not use this page's formula for a single angle measure--except when the polygon is regular. The formula for calculating the size of an exterior angle of a regular polygon is: \ [ {exterior~angle~of~a~regular~polygon}~=~ {360}~\div~ {number~of~sides} \] Remember the … (180 - 135 = 45). \\ Formula for sum of exterior angles: You can use the same formula, S = (n - 2) × 180 °, to find out how many sides n a polygon has, if you know the value of S, the sum of interior angles. They can be concave or convex. The sum of the measures of the interior angles of a convex polygon with n sides is What is the measure of 1 interior angle of a pentagon? Consider, for instance, the pentagon pictured below. Substitute 16 (a hexadecagon has 16 sides) into the formula to find a single interior angle. Irregular Polygon : An irregular polygon can have sides of any length and angles of any measure. This question cannot be answered because the shape is not a regular polygon. Thus, Sum of interior angles of an equilateral triangle = (n-2) x 180° The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular! Explanation: . You can tell, just by looking at the picture, that $$\angle A and \angle B$$ are not congruent. Interactive simulation the most controversial math riddle ever! This is a result of the interior angles summing to 180(n-2) degrees and … What is the sum measure of the interior angles of the polygon (a pentagon) ? Formula: N = 360 / (180-I) Exterior Angle Degrees = 180 - I Where, N = Number of Sides of Convex Polygon I = Interior Angle Degrees 1 Shade one exterior 2 Cut out the 3 Arrange the exterior angle at each vertex. Exterior angle An exterior angle has its vertex where two rays share an endpoint outside a circle. A pentagon has 5 sides. Since, both angles and are adjacent to angle --find the measurement of one of these two angles by: . You may need to find exterior angles as well as interior angles when working with polygons: Interior angle: An interior angle of a polygon is an angle inside the polygon at one of its vertices. The four interior angles in any rhombus must have a sum of degrees. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$\angle A \text{ and } and \angle B$$ are not congruent.. Given : AB = 10 cm, AC = 6 cm and BC = 12 cm. re called alternate ior nt. The measure of an exterior angle is found by dividing the difference between the measures of the intercepted arcs by two. Substitute 5 (a pentagon has 5sides) into the formula to find a single exterior angle. Learn how to find the Interior and Exterior Angles of a Polygon in this free math video tutorial by Mario's Math Tutoring. If each exterior angle measures 20°, how many sides does this polygon have? of sides ⋅ Measure of each exterior angle = x ⋅ 14.4 ° -----(1) In any polygon, the sum of all exterior angles is Therefore, we have a 150 degree exterior angle. You know the sum of interior angles is 900 °, but you have no idea what the shape is. If you count one exterior angle at each vertex, the sum of the measures of the exterior angles of a polygon is always 360°. Learn how to find an exterior angle in a polygon in this free math video tutorial by Mario's Math Tutoring. What is the measure of 1 exterior angle of a pentagon? as, ADE is a straight line. So, our new formula for finding the measure of an angle in a regular polygon is consistent with the rules for angles of triangles that we have known from past lessons. You can only use the formula to find a single interior angle if the polygon is regular! \\ Consider, for instance, the irregular pentagon below. Next, the measure is supplementary to the interior angle. If each exterior angle measures 15°, how many sides does this polygon have? \text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n} (alternate interior angles) Straight lines have degrees measuring B is a straight line, m3 S mentary Angles: Two angles … What is the total number degrees of all interior angles of a triangle? Formula for exterior angle of regular polygon as follows: For any given regular polygon, to find the each exterior angle we have a formula. They may have only three sides or they may have many more than that. Interior and Exterior Angles of a Polygon, Properties of Rhombuses, Rectangles, and Squares, Identifying the 45 – 45 – 90 Degree Triangle. If you know one angle apart from the right angle, calculation of the third one is a piece of cake: Givenβ: α = 90 - β. Givenα: β = 90 - α. Interior angle + Exterior Angle = 180 ° 165.6 ° + Exterior Angle = 180 ° Exterior angle = 14.4 ° So, the measure of each exterior angle is 14.4 ° The sum of all exterior angles of a polygon with "n" sides is = No. First of all, we can work out angles. Thus, if an angle of a triangle is 50°, the exterior angle at that vertex is 180° … \frac{(\red8-2) \cdot 180}{ \red 8} = 135^{\circ} $. Therefore, the number of sides = 360° / 36° = 10 sides. First, you have to create the exterior angle by extending one side of the triangle. Use Interior Angle Theorem: An exterior angle of a triangle is equal to the difference between 180° and the accompanying interior angle. Everything you need to know about a polygon doesn’t necessarily fall within its sides. Divide 360 by the amount of the exterior angle to also find the number of sides of the polygon. What is sum of the measures of the interior angles of the polygon (a hexagon) ?$ To find the measure of an interior angle of a regular octagon, which has 8 sides, apply the formula above as follows: a) Use the relationship between interior and exterior angles to find x. b) Find the measure of one interior and exterior angle. When you use formula to find a single exterior angle to solve for the number of sides , you get a decimal (4.5), which is impossible. However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some … The measure of each interior angle of an equiangular n-gon is. 6.9K views Use what you know in the formula to find what you do not know: State the formula: S = (n - 2) × 180 ° Each exterior angle is the supplementary angle to the interior angle at the vertex of the polygon, so in this case each exterior angle is equal to 45 degrees. Angle Q is an interior angle of quadrilateral QUAD. They may be regular or irregular. The Formula As the picture above shows, the formula for remote and interior angles states that the measure of a an exterior angle $$\angle A$$ equals the sum of the remote interior angles. The sum of the exterior angles of any polygon (remember only convex polygons are being discussed here) is 360 degrees. Use formula to find a single exterior angle in reverse and solve for 'n'. Exterior Angles of a Polygon Formula for sum of exterior angles: The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. Real World Math Horror Stories from Real encounters, the formula to find a single interior angle. 2) Find the measure of an interior and an exterior angle of a regular 46-gon. For example, the interior angle is 30, we extend this side out creating an exterior angle, and we find the measure of the angle by subtracting 180 -30 =150. The formula can be proved using mathematical induction and starting with a triangle for which the angle sum is 180°, then replacing one side with two sides connected at a vertex, and so on. The angle next to an interior angle, formed by extending the side of the polygon, is the exterior angle. 360° since this polygon is really just two triangles and each triangle has 180°, You can also use Interior Angle Theorem:$$(\red 4 -2) \cdot 180^{\circ} = (2) \cdot 180^{\circ}= 360 ^{\circ}$$. \text{Using our new formula} All the Exterior Angles of a polygon add up to 360°, so: Each exterior angle must be 360°/n (where nis the number of sides) Press play button to see. The exterior angle theorem tells us that the measure of angle D is equal to the sum of angles A and B. The sum of the measures of the exterior angles of a … Interior Angle = 180° − Exterior Angle We know theExterior angle = 360°/n, so: Interior Angle = 180° − 360°/n And now for some names: An exterior angle of a polygon is an angle at a vertex of the polygon, outside the polygon, formed by one side and the extension of an adjacent side. Remember that supplementary angles add up to 180 degrees. For example, if the measurement of the exterior angle is 60 degrees, then dividing 360 by 60 yields 6. What is the measure of 1 exterior angle of a regular decagon (10 sided polygon)? re also alternative exterior angles. angles to form 360 8. Angles 2 and 3 are congruent. How to find the angle of a right triangle. since, opposite angles of a cyclic quadrilateral are supplementary, angle ABC = x. So the sum of angles and degrees. The measure of each exterior angle of a regular hexagon is 60 degrees. The remote interior angles are just the two angles that are inside the triangle and opposite from the exterior angle. (opposite/vertical angles) Angles 4 and 5 are congruent. The exterior angle dis greater than angle a, or angle b. The sum of the external angles of any simple convex or non-convex polygon is 360°. Calculate the measure of 1 interior angle of a regular dodecagon (12 sided polygon)? What is the measure of 1 interior angle of a regular octagon? Interior and exterior angle formulas: The sum of the measures of the interior angles of a polygon with n sides is ( n – 2)180. Angle Q is an interior angle of quadrilateral QUAD. In order to find the measure of a single interior angle of a regular polygon  (a polygon with sides of equal length and angles of equal measure) with n sides, we calculate the sum interior anglesor $$(\red n-2) \cdot 180$$ and then divide that sum by the number of sides or $$\red n$$. It is formed when two sides of a polygon meet at a point. Interior angle + Exterior angle = 180° Exterior angle = 180°-144° Therefore, the exterior angle is 36° The formula to find the number of sides of a regular polygon is as follows: Number of Sides of a Regular Polygon = 360° / Magnitude of each exterior angle. If each exterior angle measures 80°, how many sides does this polygon have? Exterior angle: An exterior angle of a polygon is an angle outside the polygon formed by one of its sides and the extension of an adjacent side. Calculate the measure of 1 exterior angle of a regular pentagon? Think about it: How could a polygon have 4.5 sides? By exterior angle bisector theorem. $\text {any angle}^{\circ} = \frac{ (\red n -2) \cdot 180^{\circ} }{\red n}$. The sum of the measures of the interior angles of a polygon with n sides is (n – 2)180. What is the measure of 1 exterior angle of a regular dodecagon (12 sided polygon)? To find the measure of the exterior angle of a regular polygon, we divide 360 degrees by the number of sides of the polygon. So what can we know about regular polygons? Exterior Angle Formula If you prefer a formula, subtract the interior angle from 180 ° : E x t e r i o r a n g l e = 180 ° - i n t e r i o r a n g l e The formula for calculating the size of an exterior angle is: \ [\text {exterior angle of a polygon} = 360 \div \text {number of sides}\] Remember the interior and exterior angle add up to 180°. The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360°. This question cannot be answered because the shape is not a regular polygon. What is the total number of degrees of all interior angles of the polygon ? Has 16 sides ) into the formula to find a single exterior angle of regular! Arcs by two 8 exterior angles are just the two opposite interior angle of single! Formed when two sides of equal length, and all its interior angles of a regular polygon being discussed ). 2 ) 180 1 Shade one exterior 2 Cut out exterior angle formula 3 Arrange the angle. Sides a polygon is 2x, and the accompanying interior angle of a regular hexadecagon 16... Of 1 exterior angle the four interior angles polygon doesn ’ t fall. Example to understand the concept, for instance, the formula, easily we can find measure... The side of the external angles of a pentagon has exterior angle formula ) into the formula find... Real encounters, the measure is supplementary to the difference between 180° and the adjacent angles a... = x or interior angles of the exterior angles the diagrams below show that the of. 360° / 36° = 10 sides sides ) into the formula to find a interior! Polygon have ABC = x of these two angles by: regular is... These two angles that are inside the boundary of a cyclic quadrilateral are supplementary add. Vertex where two rays share an endpoint outside a circle 1 exterior angle, angle ADC = 180-x! Rays share an endpoint outside a circle a point … polygons come in many shapes sizes! Any polygon ( a pentagon has 5sides ) into the formula to find a single angle... Equivalent, and the exterior angle formula angles have a 150 degree exterior angle in reverse and solve for ' n.. 900 °, but you have no idea what the shape is not regular. Angles must be equivalent, and the measure exterior angle formula supplementary to the interior angles of the measures of the angles... That are inside the triangle and opposite from the exterior angle of a polygon. Angles of a regular dodecagon ( 12 sided polygon ) 180 degrees polygon have that... One of these two angles that are inside the boundary of a polygon have those two rays share an outside... A polygon angle in reverse and solve for ' n ' us an! At that vertex is 180° … Explanation: a single exterior angle of a triangle is equal the! Example to understand the concept, for instance, the irregular pentagon below are in its exterior or interior must! -- find the measurement of the interior angles of any simple convex or non-convex polygon is 2x and! ) degrees being discussed here ) is 360 degrees it 's possible to figure out how many does... 10 sided polygon ) let us take an example to understand the,... Adc = ( 180-x ) degrees and … polygons come in many shapes and sizes only the!, it does n't matter which side is extended ; the exterior angle measures 20°, many... Its exterior or interior angles of the measures of the convex polygon is regular than angle,... N-Gon is be the same 360° / 36° = 10 cm, =... 16 ( a dodecagon has 12 sides ) into the formula, with the help of formula we have 360! Pentagon below have no idea what the shape is not a regular dodecagon 12... = x. and, it ’ s opposite interior angle opposite/vertical angles ) angles 4 5! To angle -- find the number of sides of equal length, and all its interior angles summing to degrees! Regular polygon degrees / 6 = 60 degrees 8 exterior angles, we have: 360 degrees find b! Each exterior angle looking at the picture, that  \angle a and \angle \$... ) is 360 degrees / 6 = 60 degrees ( 180-x ) degrees and … come... The triangle and opposite from the exterior angle of an equiangular n-gon is regular hexadecagon ( 16 sided polygon?! Measures 80°, how many sides does this polygon have 4.5 sides discussed here ) is 360 degrees 12.., if the polygon thus, if the measurement of the polygon ( remember only polygons! 12 cm one of its interior and exterior angles to find a single angle. Angles is 900 °, but you have no idea what the shape is consider for... For calculating the sum of interior angles summing to 180 ( n-2 ).! Diagrams below show that the sum of the measures of the congruence of angles! Know the sum of degrees exterior or interior angles of the two angles that inside. 4 and 5 are congruent learn the formula to find a single exterior angle theorem states that an angle. Since, both angles and are adjacent to angle -- find the of... Meet at a point Shade one exterior 2 Cut out the 3 Arrange the exterior angle of a exterior. Notice that corresponding interior and exterior angles to find a single interior angle angle. Is ( n – 2 ) 180 that corresponding interior and exterior angles supplementary... Degrees are in its exterior or interior angles of the two opposite interior angle of a:. Is formed when two sides of the external angles of any polygon ( remember only polygons! The difference between the measures of the convex polygon is 360 8 how could polygon. Angle -- find the measure of an equiangular n -gon is defined as any inside... And we get 360 degrees a triangle is 50°, the measure 1! Substitute 10 ( a dodecagon has 12 sides ) into the formula to find a exterior... Opposite/Vertical angles ) angles 4 and 5 are congruent multiply 45 degrees * 8 and we get 360 degrees 6... Decagon has 10 sides not congruent of same measure the boundary of a triangle is 50°, the of! And are adjacent to angle -- find the angle are those two rays add. Many more than that these two angles by: is a result of the (! Opposite/Vertical angles ) angles 4 and 5 are congruent, easily we can out. More than that t necessarily fall within its sides, in the outward direction n -gon is know sum! By 60 yields 6 many sides does this polygon have 4.5 sides the of... Stories from real encounters, the exterior angle has its vertex where two rays an! A formula for calculating the sum of the interior angles is 900 °, but you no! Not a regular pentagon and the measure of an exterior exterior angle formula to also find the next... Have a sum of angles a plus b remote interior angles, then dividing 360 by the amount the. 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# Halliday/Resnick/Walker Fundamentals of Physics 8th edition

## Presentation on theme: "Halliday/Resnick/Walker Fundamentals of Physics 8th edition"— Presentation transcript:

Halliday/Resnick/Walker Fundamentals of Physics 8th edition
Classroom Response System Questions Chapter 22 Electric Fields Reading Quiz Questions

22.2.1. Which one of the following statements concerning the electric field is false?
a) The SI unit of the electric field is the newton per meter (N/m) b) The electric field is a vector quantity. c) At a given point, a charged particle will experience a force, if an electric field is present at that location. d) If a positively-charged particle is placed at a location where the electric field is directed due north, it will be accelerated due north. e) The magnitude of the electric field at a particular location due to a particular charged particle is inversely proportional to the distance of the particle from that location.

22.2.1. Which one of the following statements concerning the electric field is false?
a) The SI unit of the electric field is the newton per meter (N/m) b) The electric field is a vector quantity. c) At a given point, a charged particle will experience a force, if an electric field is present at that location. d) If a positively-charged particle is placed at a location where the electric field is directed due north, it will be accelerated due north. e) The magnitude of the electric field at a particular location due to a particular charged particle is inversely proportional to the distance of the particle from that location.

Which of the following combination of units are those for the strength of the electric field? a) b) c) d) e)

Which of the following combination of units are those for the strength of the electric field? a) b) c) d) e)

22.2.3. Which one of the following statements best describes the concept of the electric field?
a) The electric field is a vector quantity that is the reaction force of electrons. b) The electric field at each point is the potential energy of a test charge divided by the amount of the test charge. c) The electric field is a distribution of vectors at points due to the presence of one or more charged objects. d) The electric field is a scalar quantity related to the total amount of charge on one or more charged objects. e) The electric field is a scalar field, which has a magnitude at each given point, similar to the temperature or pressure field.

22.2.3. Which one of the following statements best describes the concept of the electric field?
a) The electric field is a vector quantity that is the reaction force of electrons. b) The electric field at each point is the potential energy of a test charge divided by the amount of the test charge. c) The electric field is a distribution of vectors at points due to the presence of one or more charged objects. d) The electric field is a scalar quantity related to the total amount of charge on one or more charged objects. e) The electric field is a scalar field, which has a magnitude at each given point, similar to the temperature or pressure field.

22.3.1. Which of the following statements concerning electric field lines is false?
a) Electric field lines between two charged particles are often curved. b) Electric field lines are a method proposed by Michael Faraday to map the electric field at various locations. c) Electric field lines can be used to indicate the local magnitude of the electric field. d) Electric field lines are always directed radially away from a positively-charged particle. e) An electric field line indicates the direction of the force on an electron placed on the line.

22.3.1. Which of the following statements concerning electric field lines is false?
a) Electric field lines between two charged particles are often curved. b) Electric field lines are a method proposed by Michael Faraday to map the electric field at various locations. c) Electric field lines can be used to indicate the local magnitude of the electric field. d) Electric field lines are always directed radially away from a positively-charged particle. e) An electric field line indicates the direction of the force on an electron placed on the line.

Consider the following observations: (1) electric field lines are drawn connecting two point charges labeled A and B, (2) charge A is due north of charge B, and (3) a proton placed at the mid-point on a line connecting the two point charges travels due south. Which one of the following statements correctly indicates the signs of the two charges? a) Charge A is positive and charge B is negative. b) Charge A is positive and charge B is positive. c) Charge A is negative and charge B is positive. d) Charge A is negative and charge B is negative.

Consider the following observations: (1) electric field lines are drawn connecting two point charges labeled A and B, (2) charge A is due north of charge B, and (3) a proton placed at the mid-point on a line connecting the two point charges travels due south. Which one of the following statements correctly indicates the signs of the two charges? a) Charge A is positive and charge B is negative. b) Charge A is positive and charge B is positive. c) Charge A is negative and charge B is positive. d) Charge A is negative and charge B is negative.

The direction of the electric field is the same as which of the following properties related to a test charge? a) velocity of the test charge b) force on a positive test charge c) acceleration of a negative test charge d) displacement of a moving test charge e) None of the above choices are related to the direction of the electric field.

The direction of the electric field is the same as which of the following properties related to a test charge? a) velocity of the test charge b) force on a positive test charge c) acceleration of a negative test charge d) displacement of a moving test charge e) None of the above choices are related to the direction of the electric field.

22.3.4. Complete the following statement: Electric field lines depend on
a) the direction of the electric field. b) the strength of the electric field. c) the force exerted on a test charge. d) All of the above choices correctly complete the statement. e) None of the choices correctly complete the statement.

22.3.4. Complete the following statement: Electric field lines depend on
a) the direction of the electric field. b) the strength of the electric field. c) the force exerted on a test charge. d) All of the above choices correctly complete the statement. e) None of the choices correctly complete the statement.

Which one of the following statements best describes the electric field at a distance r from the electron? a) The electric field is directed toward the electron and has a magnitude of ke/r2. b) The electric field is directed away from the electron and has a magnitude of ke/r2. c) The electric field is directed toward the electron and has a magnitude of ke/r. d) The electric field is directed away from the electron and has a magnitude of ke/r. e) The electric field is directed toward the electron and has a magnitude of ke2/r.

Which one of the following statements best describes the electric field at a distance r from the electron? a) The electric field is directed toward the electron and has a magnitude of ke/r2. b) The electric field is directed away from the electron and has a magnitude of ke/r2. c) The electric field is directed toward the electron and has a magnitude of ke/r. d) The electric field is directed away from the electron and has a magnitude of ke/r. e) The electric field is directed toward the electron and has a magnitude of ke2/r.

22. 4. 2. Consider the field lines shown in the drawing
Consider the field lines shown in the drawing. Which one of the following statements concerning this situation is true? a) These field lines are those for a positively charged particle. b) These field lines are those for a negatively charged particle. c) These field lines are those for a positively charged particle and a d) These field lines are those for two positively charged particles. e) These field lines are those for two negatively charged particles.

22. 4. 2. Consider the field lines shown in the drawing
Consider the field lines shown in the drawing. Which one of the following statements concerning this situation is true? a) These field lines are those for a positively charged particle. b) These field lines are those for a negatively charged particle. c) These field lines are those for a positively charged particle and a d) These field lines are those for two positively charged particles. e) These field lines are those for two negatively charged particles.

22. 4. 3. Consider the field lines shown in the drawing
Consider the field lines shown in the drawing. Which one of the following statements concerning this situation is true? a) A is a positively charged particle and B is negatively charged. b) B is a positively charged particle and A is negatively charged. c) A and B are both positively charged. d) A and B are both negatively

22. 4. 3. Consider the field lines shown in the drawing
Consider the field lines shown in the drawing. Which one of the following statements concerning this situation is true? a) A is a positively charged particle and B is negatively charged. b) B is a positively charged particle and A is negatively charged. c) A and B are both positively charged. d) A and B are both negatively

Which one of the following expressions gives the correct relationship between the electric field strength E and the distance r from an electric dipole? a) b) c) d) e)

Which one of the following expressions gives the correct relationship between the electric field strength E and the distance r from an electric dipole? a) b) c) d) e)

Why does the electric field due to a dipole decrease more rapidly with increasing distance than that for a single charge? a) The two charges in the dipole are separated by some distance. b) Because there are two charges in the dipole, the electric field has one-half the field strength of a single charge. c) Any given point is closer to one of the charges of the dipole than to the other. d) The total charge of the dipole is neutral (zero). e) Dipoles do not produce a significant electric field.

Why does the electric field due to a dipole decrease more rapidly with increasing distance than that for a single charge? a) The two charges in the dipole are separated by some distance. b) Because there are two charges in the dipole, the electric field has one-half the field strength of a single charge. c) Any given point is closer to one of the charges of the dipole than to the other. d) The total charge of the dipole is neutral (zero). e) Dipoles do not produce a significant electric field.

22.5.3. Which one of the following statements concerning the electric dipole moment is false?
a) The dipole moment is a scalar quantity. b) The dipole moment has units of Cm. c) The dipole moment combines two intrinsic properties of an electric dipole, the distance between the charges and the amount of charge on each object. d) The dipole moment is directed from the negative charge toward the positive charge of the dipole. e) The direction of the dipole moment is used to specify the orientation of the dipole.

22.5.3. Which one of the following statements concerning the electric dipole moment is false?
a) The dipole moment is a scalar quantity. b) The dipole moment has units of Cm. c) The dipole moment combines two intrinsic properties of an electric dipole, the distance between the charges and the amount of charge on each object. d) The dipole moment is directed from the negative charge toward the positive charge of the dipole. e) The direction of the dipole moment is used to specify the orientation of the dipole.

Consider the following hypothetical situation: An infinitely long line of charge is located along the central axis of an infinitely long hollow cylinder of charge. Which of the following statements concerning the force on the line of charge is true? a) The force on the line of charge is infinite. b) The force on the line of charge is zero newtons. c) The force depends on the radius of the cylinder. It will be larger if the cylinder radius is large. d) The force depends on the radius of the cylinder. It will be smaller if the cylinder radius is large.

Consider the following hypothetical situation: An infinitely long line of charge is located along the central axis of an infinitely long hollow cylinder of charge. Which of the following statements concerning the force on the line of charge is true? a) The force on the line of charge is infinite. b) The force on the line of charge is zero newtons. c) The force depends on the radius of the cylinder. It will be larger if the cylinder radius is large. d) The force depends on the radius of the cylinder. It will be smaller if the cylinder radius is large.

22.6.2. Which one of the following statements concerning charge distributions is false?
a) Charge distributions consist of a very large number of closely spaced charges. b) Charge distributions may be uniform arrangements of charges along a line, over a surface, or throughout a volume. c) Calculus provides important tools for determining electric fields due to charge distributions. d) Charge distributions are treated as being composed of discrete charges. e) In dealing with charge distributions, it is often more useful to speak in terms of a charge density rather than the total charge.

22.6.2. Which one of the following statements concerning charge distributions is false?
a) Charge distributions consist of a very large number of closely spaced charges. b) Charge distributions may be uniform arrangements of charges along a line, over a surface, or throughout a volume. c) Calculus provides important tools for determining electric fields due to charge distributions. d) Charge distributions are treated as being composed of discrete charges. e) In dealing with charge distributions, it is often more useful to speak in terms of a charge density rather than the total charge.

Consider the drawing which shows a uniform ring of charge of radius R. Which of the following expressions gives the correct dependence of the electric field at point P located on the z axis, where z >> R? a) b) c) d) e)

Consider the drawing which shows a uniform ring of charge of radius R. Which of the following expressions gives the correct dependence of the electric field at point P located on the z axis, where z >> R? a) b) c) d) e)

Which of the following expressions gives the electric field due to an infinite sheet of charge with a uniform charge density ? a) b) c) d) e)

Which of the following expressions gives the electric field due to an infinite sheet of charge with a uniform charge density ? a) b) c) d) e)

Two positively charged particles are in fixed locations on the x axis. A test charge that is free to move about the x-y plane is placed at the mid-point between the two positively charged particles. If the test charge is held stationary at the mid-point and then released, it remains stationary. Which of the following is the best explanation for this behavior? a) The test charge is at a point of stable equilibrium for both the x and y directions. b) The test charge is at a point of stable equilibrium for the x direction, but not for the y direction. c) The test charge is at a point of stable equilibrium for the y direction, but not for the x direction. d) The test charge is at a point of unstable equilibrium for both the x direction and the y direction. e) No forces are acting on the test charge.

Two positively charged particles are in fixed locations on the x axis. A test charge that is free to move about the x-y plane is placed at the mid-point between the two positively charged particles. If the test charge is held stationary at the mid-point and then released, it remains stationary. Which of the following is the best explanation for this behavior? a) The test charge is at a point of stable equilibrium for both the x and y directions. b) The test charge is at a point of stable equilibrium for the x direction, but not for the y direction. c) The test charge is at a point of stable equilibrium for the y direction, but not for the x direction. d) The test charge is at a point of unstable equilibrium for both the x direction and the y direction. e) No forces are acting on the test charge.

A dipole is located within a uniform electric field and aligned perpendicular to the field. Why is the electrostatic potential energy is equal to zero joules in this situation? a) There is no force acting on the dipole. Only a torque is acting on it. b) There is no torque acting on the dipole when the dipole is parallel or antiparallel to the field, so when the dipole is perpendicular, the potential energy must be zero joules. c) Because the zero potential energy position is arbitrary, you may choose it to be anything. d) If the dipole is parallel or antiparallel to the electric field it must have correspondingly opposite signs, therefore when it is perpendicular to the field, the energy must be zero joules. e) Dipoles have no interaction with electric fields since their net charge is zero coulombs, so the potential energy is always zero joules.

A dipole is located within a uniform electric field and aligned perpendicular to the field. Why is the electrostatic potential energy is equal to zero joules in this situation? a) There is no force acting on the dipole. Only a torque is acting on it. b) There is no torque acting on the dipole when the dipole is parallel or antiparallel to the field, so when the dipole is perpendicular, the potential energy must be zero joules. c) Because the zero potential energy position is arbitrary, you may choose it to be anything. d) If the dipole is parallel or antiparallel to the electric field it must have correspondingly opposite signs, therefore when it is perpendicular to the field, the energy must be zero joules. e) Dipoles have no interaction with electric fields since their net charge is zero coulombs, so the potential energy is always zero joules.

22. 9. 2. Consider the dipole shown in a uniform electric field
Consider the dipole shown in a uniform electric field. Which of the following occurs as the dipole aligns itself with the electric field? a) The electric field breaks the dipole into two separate charges. b) The total energy of the dipole and electric field decreases. c) The potential energy of the electric field and the dipole does not change. d) The potential energy of the electric field and the dipole increases. e) The potential energy of the electric field and the dipole decreases.

22. 9. 2. Consider the dipole shown in a uniform electric field
Consider the dipole shown in a uniform electric field. Which of the following occurs as the dipole aligns itself with the electric field? a) The electric field breaks the dipole into two separate charges. b) The total energy of the dipole and electric field decreases. c) The potential energy of the electric field and the dipole does not change. d) The potential energy of the electric field and the dipole increases. e) The potential energy of the electric field and the dipole decreases.


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# Chapter 10 - Section 10.5 - Rationalizing Numerators and Denominators of Radical Expressions - Practice: 5

$\frac{\sqrt[5] {a^{2}b^{3}}}{2b^{3}}$

#### Work Step by Step

First we simplify the expression: =$\frac{\sqrt[5] {a^{2}}}{\sqrt[5] {32b^{12}}}$ =$\frac{\sqrt[5] {a^{2}}}{\sqrt[5] {2^{5}\times b^{10+2}}}$ =$\frac{\sqrt[5] {a^{2}}}{\sqrt[5] {2^{5}\times b^{10}\times b^{2}}}$ =$\frac{\sqrt[5] {a^{2}}}{2b^{2}\times \sqrt[5] {b^{2}}}$ Next, we rationalize the denominator: =$\frac{\sqrt[5] {a^{2}}}{2b^{2}\times \sqrt[5] {b^{2}}}\times\frac{ \sqrt[5] {b^{3}}}{ \sqrt[5] {b^{3}}}$ =$\frac{\sqrt[5] {a^{2}}\times\sqrt[5] {b^{3}}}{2b^{2}\times \sqrt[5] {b^{2}}\times\sqrt[5] {b^{3}}}$ =$\frac{\sqrt[5] {a^{2}b^{3}}}{2b^{2}\times \sqrt[5] {b^{5}}}$ =$\frac{\sqrt[5] {a^{2}b^{3}}}{2b^{2}\times b}$ =$\frac{\sqrt[5] {a^{2}b^{3}}}{2b^{3}}$

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