   Chapter 4.7, Problem 20E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# (a) Show that of all the rectangles with a given area, the one with smallest perimeter is a square.(b) Show that of all the rectangles with a given perimeter, the one with greatest area is a square.

(a)

To determine

To show: The rectangle with the smallest perimeter is a square when the area is given.

Explanation

Given:

The area of the rectangle is given.

Proof:

Let the length and width of the rectangle be x and y.

Then the area is,

A=xyy=Ax

The perimeter of the rectangle is P=2x+2y .

Substitute y=Ax in P=2x+2y ,

P=2x+2Ax

Differentiate P with respect to x,

dPdx=22Ax2

For critical points, dPdx=0 .

22Ax2=0x2=Ax=±A=A, since area is always positive

Differentiate dPdx with respect to x,

(b)

To determine

To show: The rectangle with the greatest area is a square when the perimeter is given.

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Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th 