Minimizing perimeter of a stadium

Karol

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Joined
May 8, 2016
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2
Hello,

I have a problem that drives me insane. Here it is:

There is a stadium, built out of three pieces: a rectangle and two semicircles. A is the area of the stadium and the two semicircles we call the shaded area. Prove that the perimeter of the stadium is minimized when the shaded area is 2/3 A.

here is how I tried to solve it. The problem is to minimize perimeter of a stadium of a fixed area, therefore I assumed A=1. I called the radius r, and the straight sides a. Therefore area A = pi r^2 + 2 pi r = 1, therefore a = (1 - pi r^2)/2r. The shaded area is S = pi r^2. The perimeter is P = 2 pi r + 2a. Substituting for a, there is P = pi r + 1/r. Differentiating, the minimum is at r = 1/sqrt(pi), so a=0 and S=1, which is incorrect. Where am I making a mistake?

please help me,
Karol
 
Hello,


I have a problem that drives me insane. Here it is:


There is a stadium, built out of three pieces: a rectangle and two semicircles. A is the area of the stadium and the two semicircles we call the shaded area. Prove that the perimeter of the stadium is minimized when the shaded area is 2/3 A.


here is how I tried to solve it. The problem is to minimize perimeter of a stadium of a fixed area, therefore I assumed A=1. I called the radius r, and the straight sides a. Therefore area A = pi r^2 + 2 pi r = 1, therefore a = (1 - pi r^2)/2r. The shaded area is S = pi r^2. The perimeter is P = 2 pi r + 2a. Substituting for a, there is P = pi r + 1/r. Differentiating, the minimum is at r = 1/sqrt(pi), so a=0 and S=1, which is incorrect. Where am I making a mistake?


please help me,
Karol
 
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