College Algebra, Chapter 4, 4.1, Section 4.1, Problem 76

Suppose that a rain gutter is formed by bending up the sides of a 30 inch wide rectangular metal sheet as shown in the figure below.

"Please refer to the figure in the book"

a.) Find the function that models the cross sectional area of the gutter in terms of $x$.

b.) Find the value of $x$ that maximizes the cross sectional area of the gutter in terms of $x$.

c.) What is the maximum cross sectional area of the gutter?

a.) By observation, the cross sectional area of the gutter is $A = x(30 - 2x)$

$A = 30x - 2x^2$

b.) The cross sectional area is a quadratic function with $a = -2$ and $b = 30$. Thus, its maximum value occurs when

$\displaystyle x = \frac{-b}{2a} = \frac{-30}{2(-2)} = \frac{30}{4} = \frac{15}{2}$ inches

c.) Therefore, the maximum value of area is

$\displaystyle A = 30x - 2x^2 = 30 \left( \frac{15}{2} \right) - 2 \left( \frac{15}{2} \right)^2 = \frac{225}{2} \text{ in}^2$