Posted by **sunny** on Wednesday, May 8, 2013 at 1:03am.

Given that a sports arena will have a 1400 meter perimeter and will have semi-

circles at the ends with a possible rectangular area between the semi-circles, determine

the dimensions of the rectangle and semi-circles that will maximize the total area.

- math -
**Reiny**, Wednesday, May 8, 2013 at 6:54am
Let the radius of the end sem-circles be r

making the width of the rectangle to be 2r

let the length of the rectangle be x

The perimeter of the field is 2x + 2πr

= 1400

x + πr = 700

x = 700-πr

area = πr^2 + 2rx

= πr^2 + 2r(700-πr)

= πr^2 + 1400r - 2πr^2

d(area)/dr = 2πr + 1400 - 4πr = 0 for a max of area

1400 - 2πr = 0

2πr = 1400

r = 700/π

then x = 700 - π(700/π) = 0

Unexpected strange result!

BUT there is nothing wrong with the calculations, it is the wording of the question that is flawed.

Of course the largest area of any region with a given perimeter is a circle, which my solution has shown.

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