   Chapter 4.7, Problem 79E ### 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 maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W. [Hint: Express the area as a function of an angle θ.]

To determine

To find: The maximum area of a rectangle that can be circumscribed about a given rectangle with length L and width W.

Explanation

Calculation:

The length of the given rectangle is L and the width of the given rectangle is W.

In Figure 1,

The rectangle FBND is inscribed inside a rectangle AOME.

The rectangle FBND touched the rectangle AOME at F,B,N,D points.

Let ABF=θ

Hence, EFD=θ

In the triangle ΔAFB

sinθ=aWa=Wsinθ

cosθ=cWc=Wcosθ

In the triangle ΔEFD

cosθ=bLb=Lcosθ

sinθ=dLd=Lsinθ

The length of the rectangle AOME is (a+b)=(Wsinθ+Lcosθ)

The width of the rectangle AOME is (c+d)=(Wcosθ+Lsinθ)

Hence, the area of the rectangle AOME,

A=(a+b)×(c+d)=(Wsinθ+Lcosθ)×(Wcosθ+Lsinθ)=W2sinθcosθ+WLsin2θ+WLcos2θ+L2sinθcosθ=W

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