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(1 polnt) In 2004,It was reported that "the relationship between body mass M and standard metabolic ratel B; among Ilving organisms remains controversiall How...

Question

(1 polnt) In 2004,It was reported that "the relationship between body mass M and standard metabolic ratel B; among Ilving organisms remains controversiall However in many cases B Is approximately proportlonal t0 the three-quarters power 0f M:K(a) Write a function that represents tnis relationship If you must Introduce a new wariable use k _(b) The average mass of an African forest elephant Is 4.7 metrilc tons and that of a typical mouse Is 27 grams Use pant (a) to determine how many tlmes

(1 polnt) In 2004,It was reported that "the relationship between body mass M and standard metabolic ratel B; among Ilving organisms remains controversiall However in many cases B Is approximately proportlonal t0 the three-quarters power 0f M: K(a) Write a function that represents tnis relationship If you must Introduce a new wariable use k _ (b) The average mass of an African forest elephant Is 4.7 metrilc tons and that of a typical mouse Is 27 grams Use pant (a) to determine how many tlmes greater Ithe metabolic rate of an elephant Is than that of & mouse. Recall that metric ton MLOO0,000 grams (times greater



Answers

Metabolic Rate The average daily metabolic rate for captive animals from weasels to elk can be expressed as a function of mass by

$r=140.2 m^{0.75}$

where $m$ is the mass of the animal (in kilograms) and $r$ is the metabolic rate (in kcal per day). Source: Wildlife Feeding and Nutrition.

(a) Suppose that the mass of a weasel is changing with respect to time at a rate $d m / d t .$ Find $d r / d t$ .
(b) Determine $d r / d t$ for a $250-\mathrm{kg}$ elk that is gaining mass at a rate of 2 $\mathrm{kg}$ per day.

In this problem were given Keebler's law, which states that, uh, calorie consumption of mammals are proportional to its body weight phrase to the points 75 power. So we can write that as calories able to some constant times body weight, shul years mass. Well, I guess as a £10. So we use Ah, I guess w for weight W to the three forts, which is the equivalent of 30.75 So this is gonna be our our function model. Keebler's law. Then the problem states. I guess I wants us to craft it so quick. So here's the graph and this is for a constant K with one. Get the player on. It was for Kagel one. So pretty much the why value it's X equals one is gonna equal K because it's scaling by a factor of K. All right, here's our plot. And now the problem wants to know if human weighing £150 needs to consume modern our 1800 calories a day. We want to estimate the calorie requirement for a horse weighing £700 a rabbit wing £9. So the calorie consumption of horse is going to be. We don't have that. So if we if we look at a human, so the human way is £150 consumes 1800 calories per day, so have K times 150 to the 3/4 power. So have 8 1500 It's divided by 1 50 to the 3/4 is equal. Okay, that'll be about that's gonna be about 41 point kind 95 um, six. So And if you want to calculate a calorie consumption of horse that weighs 700 we would have RK times 700 soothe the 3/4 power 20 points that in. So I get us 5715.1 for 59 And then we want to find the consumption for a rabbit that weighs £9 que times nine to the 3/4. One moment we'll have 218.2155 calories, all right, And finally, the question wants to know on a per pound basis of switch animal requires more calories amounts or an elephant. So since we assume that amounts weighs less than an elephant and we want to look at our graph. So her pound since our X axis, his pounds and our Y axis is calories we're gonna be looking for Where on the graph is the slope higher was that slope is going to be the change in calories over the change in pounds. And if you notice that the smaller the weight is, the greater the slope of our tangents are going to be so that would tell us that a mouse is gonna have more her pounds increase of calorie consumption than an elephant.

Yeah. So if the p M. R is given by this constant a times mass raised to the 3/4, what are the units of this constant? A well, BMR is units of power. So units of power is in um energy per second. In si units, usually jewels per seconds, which is watts. So I only have to do is take the unit for mass, which is kilograms. Move that over here and we get the units for a usually given like this units for a is jules per second times kilograms To the 3/4. So if we have a person that has an A. Of 3.4 and their way 75 kg, What's the BMR? Well, we just use the equation right there, right, With the 3.4 75-3/4. Because these are also units, kg for masses, etc. Watts for power is etc. We put those together and we get About 86.65 watts. Or you can just do a w for what's you don't need to write it up. So what is the A value? If the BMR, It's 460 watts. And the mass is 700 kg for this bear. We do 460 Divided by 700 race to the 3/4 And we get an a of about 3.38 and last little bit here. Yeah, Is a two parter. If we have 180 kg Gorilla With a BMW of 170 watts. What is the BMW of a £1,000 gorilla. Well, the first thing is we got to find out what the A value is, Right? So let's do 170 watts, Divided by 180 kg raised to the 3/4. It will get an A. Of about 3.46. Now the next step is very easy. We have the masses 1000 kg. We have are a value and we need our BMR. So our BMR is this 3.46? We just solve for times 1000 raise to the three quarters, and we should get a B. M. Are about 615.3 watts.

We're given the formula. R is equal to 140.2 m raised to the 0.75 power. The first thing we're asked is what is drd T? Okay, so we're just gonna take the derivative of both sides. We haven't been given any other information. Just what is drd T So DRD t is equal to 100 or 5.15. Um, raised to the negative 0.25 power times. D m d t. Okay. No, we want to know. Okay. What d already t is when um is equal to 250 and d m d t is equal to two. So I'm gonna take my derivative formula here and solving for DRD t. And I'm gonna plug in for em and g m d t. So, um is 250. This is still raised to the negative 0.25 power and d m d t is too. You go ahead and put that into my calculator and I'm gonna end up with Do you already t is equal to 52.89

So this phone is about caloric expenditure. Over here, I have a function of the V where f of M and B is the energy use in units of a kilocalories per hour. M is mass in grams and B is the velocity or speeding kilometers per hour. So this room has three parts to part a party wants you to find f of 300. I'm intensely wants you to find the energy use for some animal that is 300 as 300 grams and travels at 10 kilometers an hour. So all we have to do is plug in the values into the equation. So we have 25.92 tires, 300 to your 0.16 power plus. All right, better eat your point. 68 pause. 3.62 times, 300 to 0.75 in the numerator, and 10 into the denominator. And if we solve that, you get approximately 1279 kilocalories per hour kilocalories per hour. All right, so this is your answer for a part. A part b part B ask you to find the parts of the of it all the function at the 0.300 10 seller. To find this, we're first going to need it a part of the video with respect to Adam. So doing so we'll get, um equals two 25.9 soon, because I was a constant times. We have to bring down 0.68 using chronic Role and Teoh Negative 0.32 plus plus in a fraction of it with the new area. 3.62 62 time 0.75 Product ruled times unto the natives. Negatives Europe on 25 in the numerator and the denominator is just be because he is constant with respect to them. So if you plug in the point I thought following the 0.300 300 10 into our parts of derivative, we'll get approximately. We got approximately 2.906 kilocalories kilocalories per hour, her ground because we're taking the derivative with respect mass. So this is part of B, and it also this from awful. Ask you to. I interpret this value of Obama at 310. So if we were Tonto interpret this course is elevated. It would be instantaneous. Instantaneous rate change both changed all the energy for, uh, 300 ground 300 grand animal traveling at and Molitor's. An hour is approximately 2.9906 locales for our grand. So this is your interpretation for part of their of it is with respect to, um I reckon, right, That's three for 300. All right, so that is part B and part c Ask you for it. Ask you if a mouse could run at the same speed of an elephant. Ah, what? Which one way, extend more energy? Well, looking at our formula, we you can obviously tell that an elephant elefant it extends more energy because it has a greater mass, then a mouse. I also ask you, how could partner is used be used to explore this question? Well, part of the route is they can be used to explore the rate of change of the energy uses for both animals, since that's what the part of the river is. So he's


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