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A model for the surface area of a human body is given by the function 0425 0,725 S = f(w,h) =0.109lw where W is the weight (in pounds) , h is the height (in inches)...

Question

A model for the surface area of a human body is given by the function 0425 0,725 S = f(w,h) =0.109lw where W is the weight (in pounds) , h is the height (in inches) ad $' is the surface area (in square feet)8S(160,70) ad 8S(160,70) _ (a): Calculate ow dh (b). Interpret the partial derivatives obtained in G(a)

A model for the surface area of a human body is given by the function 0425 0,725 S = f(w,h) =0.109lw where W is the weight (in pounds) , h is the height (in inches) ad $' is the surface area (in square feet) 8S(160,70) ad 8S(160,70) _ (a): Calculate ow dh (b). Interpret the partial derivatives obtained in G(a)



Answers

Body surface area A model for the surface area of a human body is given by the function
$$S=f(w, h)=0.1091 w^{0.425} h^{0.725}$$
where $w$ is the weight (in pounds), $h$ is the height (in
inches), and $S$ is measured in square feet. Calculate and
interpret the partial derivatives.
(a) $$\frac{\partial S}{\partial w}(160,70) \quad \text { (b) } \frac{\partial S}{\partial h}(160,70)$$

We have a function here that models the surface area of the human body and W. Is the weight and H. Is the height. So once the partial derivative is first with respect to wait then with respect to height. So let's give that a try. So D. S. D. W. Equals Okay the 0.10. 9 1. That's a constant W. Is the thing changing? So it's derivative is 0.4 to five. W. To the 50.4 to 5 minus one. So point- .575. And then H. Is just a constant. So it just stays on their .7-5. And then D. S. D. H. Same thing. The 0.191 is a constant. This time W. is a constant. We have to take the derivative of H. so .7-5. H. To the- .275. Okay so then it wants you to plug 1 70 and 60 in. Um See if I can do that D. S. D. W. And the function uh hopefully goes weight height. So 1 70 is the W. And 60 is the H. What's up? So .10. 9 1 times .4- five. Um h. 60 to the .7-5. And then the other ones do a negative power. So I'm gonna put it on the bottom. Okay I'm just using my phone for a calculator so it's not that great. Okay so I'm gonna do the top first I have 60 to the .725 Times .4-5 Times .10.9 1 divided by 1 72. The .575. So I got .047. But I don't know if you didn't get that tried again. Okay and then you plug in the same thing for D. S. D. H. 1 74. W. And 64. H. Okay. Now the interesting part is what is it that you just found? Well s. S surface area and w. Is weight in pounds. So what you found is the rate of change of the surface area with respect to the change in pounds. So the heavier you are the more surface area you will have and it will change with with this uh Great .047 and then it will be surface area. The units will be whatever the surface area was in over pound. Um I didn't pay attention. So if it was an inches squared you'll be inches squared per pound. So every time you gain a pound then you gain point oh 47 square inches of surface area. Okay and then D. S. D. H. Would be the change in surface area with respect to the change in height. So that one would be point for whatever whatever it comes out to be into square per um inch. Okay so every time you get an inch taller you will gain this much surface area. Okay hope that

All right, and this problem were given a function of two variables. Weight and height. Determine the square foot surface area of a human body. Approximately were provided with this equation and then asked to find F of 1 60 70 which is the square feet, approximately in surface area of an individual who weighs £160 and the 70 inches tall as of 1 60 70. Using this formula is equal to one oh 91 times 1 60 to the power of 0.4 to 5. Time. 70 to the power of 700.7 to 5 is equal to 20 0.52 square feet. All right, the second part of this problem asks you to find your own surface area. Um, given that my surface area will different from anyone else's surface area, I'm going to be using the surface area of an individual who weighs £150 and is 5 ft or 60 inches tall. Could be finding f of 1 50 16 using the same exact formula plugged in the same exact way with 1 50 in place of 1, 60 and 16 places 70 we returned 17.86 square feet

This question after asked us to figure out after 1 60 comma 70 and then figure out our own surface area off of 1 60 comma. 70 Means were simply gonna be plugging into the s equation. Zero point 1091 times 1 62 0.4 to 5. Time 70 to the power of 0.7 to 5 gives us 21.7 square feet. In other words, we can write this as feet squared and then we're looking at part be part B, which asks us to figure out our own surface area Well, and this context were given 72 inches in £177. So we're gonna be doing the same thing will be plugging it into the US equation. 1091 times 17720.4 25 times 72 to 0.7 to five equals 21.87 feet squared

So we have a formula for the surface area of a human body. Onda. We see that it is f w H and R model is that this is equal to zero point 10 91 Yes, W the power of 0.4 to 5 and then that's going to be no times H to the zero point 7 to 5. So with this we want to measure the find s in square feet. We know that the weight is £160. So have f of 1 60 and 70 inches. And we see that this will give us 20 um, approximately 20.52 square feet as a result, and then thistles going to be the surface area of a human body for this case. So assuming that this is a good model, we can say that a human that is £160.70 inches tall, we will approximately find their body to be 20.52 square feet. And again, this is a general estimate. Um, the only thing that we have to keep in mind is that this is a model, so it's supposed toe model. All people that measure these specifications. It won't be exact, Um, but that's how models work. Were able to predict things for a large number of people just given these mathematical models, and that's what makes him really helpful. We also see that this is a great use of multi multi variable functions because that we see that the surface area is determined by not Onley weight but also hyped. So we need to include both on bats, one really useful, uh, 11 useful application of multi variable functions.


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