 Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q. 57

Expert-verified Found in: Page 986 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Prove that a square maximizes the area of all rectangles with perimeter P.

See the step by step solution

## Step 1. Given Information.

Given a rectangle with perimeter P. Let a and b be the dimensions of the rectangle.

## Step 2. Finding the constraint.

The perimeter is the sum of all sides, which is 2a+2b.

Therefore, the constraint function is:

$g\left(a,b\right)=2a+2b.$

The function which maximize the area is:

$f\left(a,b\right)=A=ab.$

## Step 3. Using Lagrange's multiplier.

By the method of Lagrange's multiplier,

Now whatever be the value of $\lambda$, all the components of $\nabla f$ must be same.

So,

$b=a$

Hence, it is proved that the rectangle must be a square in order to have its area maximum. ### Want to see more solutions like these? 