posted by katie .
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
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)