Find the Length and width of a rectangle with a given area and minimum perimeter. Area = A square centimeters.

Question

I'm not sure how to solve for length and width if it doesn't give me the area.

This is a calculus problem, exercise 10 on page 216 of 7th edition calculus.

Answer 1

Given L•W = A then L = A/W

The perimeter is calculated as L + W + L + W = 2L + 2W

Substituting "A/W" for "L" we get an equation for perimeter in terms of "W":

P(W) = 2(A/W) + 2W or

P(W) = 2A(W^-1) + 2W

To minimize, take the derivative and set it equal to zero:

P'(W) = -2A(W^-2) + 2

-2A(W^-2) + 2 = 0

2A(W^-2) = 2

A(W^-2) = 1

A = W²

W = √A