Question

A 100 inch piece of wire is divided into 2 pieces and each piece is bent into a square. How should this be done in order of minimize the sum of the areas of the 2 squares?
a) express the sum of the areas of the squares in terms of the lengths of x and y of the 2 pieces
b) what is the constraint equation relating x and y?
c) does this problem require optimization over anopen or closed interval?
d) solve the optimization problem

Answer 1

Let one piece to shape the first square be 4x, then let the other piece be 4y
4x + 4y = 100
x+y = 25
y = 25-x

Sum of areas = x^2 + y^2
= x^2 + (25-x)^2
= 2x^2 - 50x + 625
d(Sum of areas)/dx = 4x - 50
= 0 for a max/min of the sum of the areas
4x = 50
x = 12.5

( I defined the length as 4x instead of x to avoid fractions)