Question

27 ft of wire is to be used to form an isosceles right triangle and a circle. Determine how much of the wire should be used for the circle if the total area enclosed is to be a minimum? Maximum?

Answer 1

if the triangle has side b, and the circle has circumference c, then

2b+b√2 + c = 27
so, b = (27-c)/(2+√2)

the area is

a = 1/2 b^2 + πr^2
= 1/2 ((27-c)/(2+√2))^2 + π(c/(2π))^2

This a parabola with vertex (minimum area) at

c = 27(1 - 1/(1+(3-2√2)π)) = 9.45