   Chapter 8.2, Problem 38E Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085

Solutions

Chapter
Section Elementary Geometry For College St...

7th Edition
Alexander + 2 others
ISBN: 9781337614085
Textbook Problem

Examine several rectangles, each with an area of 36 in2, and find the dimensions of the rectangle that has the smallest perimeter. What type of figure has the smallest perimeter?

To determine

To Find:

The dimensions of the rectangle that has the smallest perimeter provided its area.

Explanation

Formula Used:

The perimeter of a polygon is the sum of the lengths of all sides of the polygon.

The perimeter P=2(b+h), where b is the base length and h is the height of the rectangle.

Area of the rectangle A=bh, where b is the base length and h is the height of the rectangle.

It is given that the area of the rectangle 36 in2.

Let 'x' be the base and 'y' be the height of the rectangle. Then the area becomes

A=xy=36

y=36x(1)

Let’s find the rectangle that has smallest perimeter, given this area.

Applying the perimeter formula, P=2x+y(2)

Substitute the value of height from equation (1) in equation (2)

Px=2x+36x(3)

Let’s minimize P(x) on (0,)

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