Question 106776
The total length is 20 inches.
Let the two lengths be x and 20-x inches
Using calculus:
The area of the square is (x/4)^2 and the area of the circle is pi*radius^2.
Now the circumference = 2*pi*r and C=20-x.
Hence r=(20-x)/(2*pi)= 10/pi-x/2pi
So total area is A=(x/4)^2+pi*(10/pi-x/2pi)^2=(x^2)/16+pi*(100/(pi)^2+x^2/4pi^2-10x/pi^2)

A= x^2/16 + 100/pi + x^2/4pi - 10x/pi

Maximum or minimum values are given when dA/dx=0 and so:
 2x/16+2x/4pi - 10/pi = 0
x/8+x/2pi - 10/pi = 0
Multiply by 8pi
x*pi + 4x - 80 = 0 = 0

x*(pi + 4) - 80 = 0
x = 80/(pi + 4)
x = 80/(22/7 + 28/7) = 80/(50/7) = (8*7/5 = 56/5 = 11.2
So the lengths are 11.2 inches and 8.8 inches ANS