Question

A carpenter is building a rectangular room with a fixed perimeter of 180ft. What are the dimensions of the largest room that can be built? What is it’s area?

Answer 1

90ft x 90ft. 8100ft^2

Explanation:

Perimeter(P) = 180 = length + width

Length(L) = 180 - width

Width(W) = 180 - length

Area = length*width = length*(180-length) = 180*length - length^2

Because the sum of length and width must always be 180, when one increases, the other decreases.

The largest area will happen when the derivative of the area is equal to 0. The derivative can be found with the power rule.

Derivative of Area (slope of Area) = 180 - 2*length

180 - 2*length = 0

180 = 2*length

length = 180/2 = 90ft

When the length is 90ft and width is 90ft, the area is 8100ft^2.