   Chapter 3.7, Problem 77E

Chapter
Section
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

Explanation

1) Calculation:

To draw the rectangle that is circumscribed by the other rectangle:

Consider the triangle with sides a, c and hypotenuse W

sinθ=aW & cosθ=cW

Now, consider the triangle with sides b, d and hypotenuse L

sinθ=dL & cosθ=bL

Thus, a=Wsinθ, b=Lcosθ, c=Wcosθ d=Lsinθ

So, the area of the circumscribed rectangle is

Aθ=a+bc+d

Substitute the above values

Aθ=Wsinθ+LcosθWcosθ+Lsinθ

Aθ=W2sinθcosθ+WLsin2θ+LWcos2θ+L2sinθcosθ

Aθ=W2+L2sinθcosθ+WLsin2θ+cos2θ

Aθ=W2+L212·2sinθcosθ+WL

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Mathematical Applications for the Management, Life, and Social Sciences 