Determine the area of the rectangle with perimeter 82 inches and maximum area.

Area = 420.25 in²

Step-by-step explanation:

A rectangle is a quadrilateral (has four sides and four angles) with two pairs of opposite and parallel sides. All angles in a rectangle are 90 degrees each. Also, opposite sides are equal.

Let x represent the length of the rectangle and y represent the width of the rectangle. Hence:

Perimeter = 2(length + width)

82 = 2(x + y)

x + y = 41

x = 41 - y

The area of the rectangle (A) = length * width

A = xy

substitute x = 41 - y:

A = (41 - y)y

A = 41y - y²

The maximum area occurs at A' (dA / dy) = 0. Hence:

A' = 41 - 2y

0 = 41 - 2y

2y = 41

y = 20.5 inches

substitute y = 20.5 inches in x = 41 - y:

x = 41 - y = 41 - 20 .5

x = 20.5 inches

The area = xy = 20.5 in * 20.5 in

Area = 420.25 in²

420.25 in²

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