A 100 inch piece of string is cut into 3 pieces. One piece forms a circle, one forms a square and 1 forms an equilateral triangle. If the perimeter of the triangle is equal in length to the perimeter of the square how long should each piece of string be to minimize the total area? What is the diameter of the circle? How long is each side of the square? How long is each side of the triangle?

Answer 1

let each side of the triangle be 4x (you will see why ...)
then each side of the square is 3x (ahh, how clever)
height of the triangle is 2√3x , by Pythagorus

let the radius of the circle be r
2πr + 24x = 100
πr = 50 - 12x
r = (50-12x)/π

total area (T) = πr^2 + 9x^2 + (1/2)(4x)(2√3x)
= π(50-12x)^2/π^2 + 4√3x^2
= (2500 - 1200x + 144x^2)/π + 9x^2 + 4√3x^2
d(T)/dx = -1200/π + 288x/π + 18x + 8√3x
= 0 for a min of T

timex π
-1200 + 288x + 18πx + 8√3π x=0
x(288 + 18π + 8√3π) = 1200
x = 3.092

r = 4.104

so diameter of circle is 8.21
perimeter of rectangle is 12x = 37.106
perimeter of square is 37.106

check:
2πr + 12x + 12x = 100