A piece of rectangular sheet metal is 21 inches wide. It is to be made into a rain gutter by turning up equal edges to form parallel sides. Let x represent the length of each of the parallel sides. For what value of x will the area of
The cross section be a maximum (and thus maximize the amount of water that the gutter will hold)? If necessary,
Round to 2 decimal places.
A) 42 in.
B) 84 in.
C) 10.5 in.
D) 5.25 in.

Question

Quizplus · Accepted Answer

The area of the cross-section of the gutter, when the sides are turned up, can be represented as A = x(21 - 2x), where x is the height of the sides. To find the value of x that maximizes the area, we differentiate the area function with respect to x and set the derivative equal to zero. The derivative of A with respect to x is dA/dx = 21 - 4x. Setting this equal to zero gives 21 - 4x = 0, which solves to x = 5.25 inches. This means that turning up the edges 5.25 inches on each side will maximize the cross-sectional area of the gutter.

A Piece of Rectangular Sheet Metal Is 21 Inches Wide