Question

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.

Answer 1

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.