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The hypotenuse of an isosceles right triangle decreases in length at a rate of $4 \mathrm{m} / \mathrm{s}$a. At what rate is the area of the triangle changing when the legs are 5 m long?b. At what rate are the lengths of the legs of the triangle changing?c. At what rate is the area of the triangle changing when the area is $4 \mathrm{m}^{2} ?$

Calculus 1 / AB

Chapter 3

Derivatives

Section 11

Related Rates

Differentiation

Campbell University

Baylor University

University of Nottingham

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hi, everyone, for this problem we are given. And I saw Seles trying Go, Let's get a triangle going here best we can with a sausage. These right triangle there. So the thing about and I saw Seles, right triangle is the two legs are equal so normally to have X y and receive without have pot news. But since those are equal, I have two exes. So let's look at our area for a triangle formula that is area equals 1/2 base times height. So in this case base and Haider Exes. So we're gonna get area equals X squared on two, all right. And we are also given the rate of change of the high pot news and we're told it is decreasing at a rate of four meters her second. So that is Deasy. The T is equal to negative four meters per second. That negative is there because again we are decreasing. So are a assets to find the a d. T. When X is equal to five meters, so access equal to buy meters here. So the a t t right here. We need to differentiate this side of the equation to find that and it will differentiate. Of course, this side was do that right now. So in a d a d t is beautiful to the chain role here, I'll bring down my exponents. So that isn't Keep the 1/2 out there times two at an ex lapped over and then I've got the derivative of the inside here DX DT So unfortunately, we're given D c e ti not the x t t. So let's see if we can figure that out. So what I'm gonna do here is I'm gonna use Pythagorean theorem around That says that Z squared is equal to X squared plus another X squared in this case. So Z square is equal to two x squared b squared there. So it's clean this up a little bit. I'll take the square root of bull signs. Square root is he squared is Z and that is equal to the square root of two acts Square clean that up a little bit. This will be square root of two times X right there. So now we have an equation. Z's really exes. Let's differentiate both sides with respect to t easy. DT is evil to square root to the x d t. So now we have a DZ d team were given. I haven't the x t t that we're looking for in this equation right here. So we know this one that plug and chug here. So that is negative. Four meters per second. These people to square through the to the x DT So clean that up on move this over here and this will say that negative for over the square root of two meters per second is equal to d x d t. It's more rationalised this denominator here to get as something let's get a little more room. Well, let's continue over here. So this says that negative square root to four times four over two peoples here the x d t. All right, so let's get rid of those. So this is the X T t is equal to negative two on the square root to must not forget our things meters per second. So now now we have something we can deal with here we have an X. So let's plug and chug and figure out what the rate of change of the area is when x equals five meters So I'm gonna plug that in these twos, of course, will cancel out. So I might end up with five. Were acts times a negative to square root of two. And there's a meters here and a meters per second here to cover those two. So when I cleaned those up, I'll get negative to temps five. Negative 10. I've got a square root of two still hanging out there. Then I've got a meters meters. So that's where years I still have my seconds. So this to keep track a little bit better over here, I'm going to end up with part a d a d t is equal to negative chan in time for square root to square meters per second is art decreasing area when X equals five meters Part B, part B. They're asking us to find the X DT. The X t t actually already did work for a bath in part a and we found it right over here. Miss the recipe lto negative to square with two meters. Sure, So let's make us a little more room here Report see All right to see assets toe fine The rate of change of the area the a G t. When area is equal to board. Excuse me. An area is equal to four square meters. All right, so we head back up here to our area formula, right? The a t t. We can solve that out. You know that that's equal to X dx dy team. But in this case, we don't know the X so around at all. The way back to our original area formula right here. X squared on two and use that to find our X. So let's see. So area is people to this right again equal to x squared to We know our area is given as four square meters. So the plug that in or square meters equals x squared on to this one that's multiplied by to hear that square roots. So I get eight square meters is equal. Teoh X squared meters there we'll take the square root of both sides. So that is the square root. Eight. I'll take this. This will get rid of right here. And that is squared eight meters with equal to X at this point. So it looks like we have Ah, fairly fun one here. Right. So must keep going over here and we'll plug and chug on our D A formula and we'll see what we get. The a t t. Excuse me, 80 people to that was X times the x d t. So you know that X is square rate of eight meters and the exit d t we found in part a and it's up here in part B. That is negative. Two on two square word of two. That is meters their second. All right, So now time to clean this mess up. So negative too. Then I've got a square root of eight square rate of two. I've got a meters about another Munir's. Then I gotta per second. So these two guys run together here, so this equals negative to square root of 16 2 times eight. Another radical there that is m smirking or meter swear her second so square root of 16. No, that is four. I'll do a negative to time that or write their negative eight square meters per second and that is our rate for the A t. T. When the area is equal to four square meters

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