   Chapter 4.7, Problem 27E ### 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 an equilateral triangle of side L if one side of the rectangle lies on the base of the triangle.

To determine

To find: the dimension of the rectangle with the largest area that can be inscribed in an equilateral triangle.

Explanation

Given:

One side of the rectangle lies on the base of the triangle.

Calculation:

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

In Figure 1, the base of the equilateral triangle is L.

Hence, the base of the triangle ABC is L2x2 .

In triangle ABC , we can write,

tan60°=yL2x23=2yLxy=32(Lx)

Area of the rectangle Z=xy .

Substitute the value of y in Z ,

Z=32(Lx)x=32(Lxx2)

Differentiate Z with respect to x,

dZdx=32(L2x)

For, critical points,

dZdx=032(L2x)=0L2x=0L

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