   Chapter 4.7, Problem 25E ### 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

# Find the dimensions of the rectangle of largest area that can be inscribed in a circle of radius r.

To determine

To find: The dimension of the rectangle of largest area that can be inscribed in a circle of radius r .

Explanation

Formula used:

Differentiation of the product of two functions,

ddx(uv)=dudxv+udvdx

Where u,v are functions of x .

Pythagorean Theorem:

If c denotes the length of the hypotenuse and a and b denote the length of the other two sides of a right angle triangle, then the Pythagorean theorem can be expressed as the Pythagorean equation: a2+b2=c2

Calculation:

Letr be the radius of the circle.

The length of the rectangle = x.

The width of the rectangle = y.

In figure 1 the rectangle is inscribed in a circle of radius r.

By Pythagoras theorem,

x2+y2=(2r)2y2=4r2x2y=(4r2x2)

Area of the rectangle is A=xy

Substitute the value of y in area of rectangle A,

A=x(4r2x2)

Differentiate with respect to x ,

Simplifying further,

(4r22x2)(4r2x2) =0

4r22x2 =0

x=±r2

Length of rectangle cannot be negative

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