A rain gutter is formed by bending up the sides of a 58inwid

A rain gutter is formed by bending up the sides of a 58-in.-wide rectangular metal sheet as shown in the figure.

(a) Find a function that models the cross-sectional area of the gutter in terms of x.
A(x) =

(b) Find the value of x that maximizes the cross-sectional area of the gutter.

x =   

(c) What is the maximum cross-sectional area for the gutter?

____ in²

Solution

(a)Let the side of the rain gutter be x inches. Then the base of the rain gutter is 58 -2x inches. Thus, the cross-sectional area of the gutter, being rectangular, is A(x) = x(58-2x) = 58x -2x².

(b) For the cross-sectional area of the gutter to be maximum, we should have dA/dx = 0 and d²A/dx² should be negative. Here, dA/dx = 58-4x. Thus, if dA/dx = 0, then 58-4x = 0 or, 4x = 58 so that x = 58/4 = 14.5 inches. Also, d²A/dx² =-2 which is negative. Thus, x = 14.5 inches for maximizing the cross-sectional area of the gutter.

(c) The maximum cross-sectional area of the gutter is A(x) when x = 14.5 inches, i.e. 58*(14.5) – 2*(14.5)² =841-420.5 = 420.5 sq. inches.