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Answer Exercise 37 if one piece is bent into a square and the other into a circle.

(a) For maximum area, all the wire should be used to make circle

(b) For minimum area, 5.6 $\mathrm{m}$ should be used for square and 4.4 $\mathrm{m}$ should be used for the circle

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Campbell University

Oregon State University

Harvey Mudd College

Boston College

we're actually asked to exercise of previous exercise exercise 37 if one piece is bent into a square and the other end to a circle This is really two parts here, so we're given a certain amount cut for the square. We'll call this amount X in the amount cut for the circle. This will be 10 minus x. Yes. Now the side length of the square is X over four. Yeah, and the circumference of the well, the area of the square. It's a function of X. This is X over four squared, which is X squared over 16. On the other hand, the circumference of the circle This is the same as 10 minus X. So we have that two pi r equals 10 minus x and so are as a function of X is r equals 10 minus X over two pi. Yes, In the area of her circle was a function of our This is pi r squared, then substituting. This is pi times 10 minus x over two pi squared. Hey, son, They want gay son, This sympathize too. 10 minus X squared over four. Hi. Now the total area of our figure shall called a. This is the area of the circle plus the area of the square, so this is X squared over 16. So again, the total area is the area of the square, plus the area of the circle so X squared over 16 plus 10 minus X squared over four pi and we want to differentiate. So a prime of X is X over eight plus two before pie is one of her two pies. Times 10 minus x times. Negative one. Yes, you want to solve the equation? A prime of X equals zero and the critical points So we have. Let's see, they were breakfast X over eight equals 10 minus X over two pi and then cross multiplying and solving. For X. We get X is equal to 80/2 pi plus eight, which is approximately 56 Now we'll evaluate her area at this critical number and at the end points of the domain and we'll compare well. Our domain is from 0 to 10. This is the value sex can take. So a of zero Well, this is zero plus 10 squared over four pi, which is 100 over four pi or 25 over pie, which is approximately 7.96 On the other hand, a of our critical value, which is 80 over. Or I guess I would say 40 over pi plus four. Plug this in. This is approximately 3.5. And if you plug in 10 a of 10, this is approximately. This is 100 over 16, which is about 6.25 So from all this, it's clear that the maximum area is obtained when act equal zero. So a is maximized when X equals zero. So the answer is, all wires should be used to make the circle. Mm, yes, What? Blue rock. And to find the minimum mhm. Well, we know we also see that the area is minimized John Close when yes, X is equal to 40 over pi plus four because we already saw is about 5.6 mhm and therefore these flu right. In this case, about 5.6 m of wire should be used for the square and the remainder 10 minus 5.6, which is 4.4. Should be meters should be used for the circle. It's just