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In the following exercises, solve the given maximum and minimum problems.If an airplane is moving at velocity $v,$ the drag $D$ on the plane is $D=a v^{2}+b / v^{2},$ where $a$ and $b$ are positive constants. Find the value(s) of $v$ for which the drag is the least.

Calculus 1 / AB

Chapter 24

Applications of the Derivative

Section 7

Applied Maximum and Minimum Problems

Derivatives

Oregon State University

Harvey Mudd College

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Okay, so we need to find the maximum volume where the volume of a suitcase is gonna be. Link times with times height, right. We're also restricted by the fact that length plus with plus height, is equal to 62. So we can rewrite one of, um either length or with in terms of the other because hiatus fix. So we'll just say length is equal to 62 minus w my insulin times the height. So we get the volume is equal to will replace length with what we just found. Some 62 minus with times, height, um, times W times h So simplifying that a bit we get 62 w h minus w squared H minus. W h squared. Now, uh, here are with is our variable. So we're going to take the derivative with respect to whip so d v d w. Here. So taking the derivative with respect to W, we had 62 h minus two w h minus h squared. Remember, we're trying to optimize, so we need to set the derivative equal to zero, and we're solving for H. So we get to w H is equal to H squared minus. Sorry to W H is equal to 62 h minus H square. So divide by two h on both sides to aged about by two age here we got that W is equal to 31 minus 1/2 a tch. Okay, so then to find the link we plug in, huh? You plug in the whip here. Okay, so we get 62 minus 31 minus 1/2 each, minus h. Okay, so we get 62 minus 31. It's going to be 31 and then this is plus 1/2 a tch within minus each. So we're left with 31 minus 1/2 a tch. So that's the same as W so Alice equal to W in this case. Now, to find the volume we plug in w N um, W l So, since lengthened with are the same, it's 31 minus 1/2 a TSH squared times each. So this is our maximum volume, given a fixed hype. Now, if we want to maximize this with respect to height, so we need to find the height that maximizes this. Okay, so let's go to and you white board here. So We know that, um, volume is going to be equal to 31 minus 1/2 each, squared h. And now we want to optimize this respect a height. So we need to find the derivative of this with respect. A height. So this is going to be we'll use product rule here. Um, so first it's gonna be the, uh So the first times the derivative the second, which is one plus second times every one of the first. So two times 31 minus 1/2 a tch. Times negative. 1/2. Okay, simplifying this a bit. We have 31 minus 1/2 a TSH squared, plus our Sorry, we can, uh, minus. So the two and the 1/2 cancel out. So we have left H times 31 minus 1/2. It's okay, so Ah, we can set this equal to zero here. Um, I hope there's a type of here that should be 31 h squared equals zero. So said that equal to zero we can factor out a 31 minus h squared. Um, so we factor out 31 minus Well, 1/2 hh. Okay, so we're still not for 31 minus 1/2 age here. If a 31 um minus 1/2 a tch here and then minus each is equal to zero, so we can set either these to be equal to zero. So this is 31 minus 1/2 H is equal to zero, or here we have 31 uh, minus minus three halves. H is equal to zero. Okay, so here on the left side, first we get that H is going to be equal to 62. Okay, it's for the for the next one, right? We have three halves. H is equal to 31 or three h equals 62 or we get H is equal to 62 divided by three. Now, if we have a TSH is equal to 62. Um, for this solution here, um, if we plug in or if we try to plug in for W and H, or if we try to find the volume here. So here volume is equal to 31 minus 1/2 of 62 squared times 62. This, um, this part is 31. So we get 31 minus 31. Uh, so it's just zero. This is just all zero. So the volume zero. So that's not what we want. Um, but if we plug in each over 32 instead h over 62 over three. This will give us our massive in volume. Okay, so the dimensions are, um Well, this is the height that will give us the maximum volume and not, um, h equals 62 here. Okay. Um, okay. Um right. Not this one. Okay, so there we go. We're done.

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