Question

In the following exercises, solve. Rounding answers to the nearest tenth.

A daycare facility is enclosing a rectangular area along the side of their building for the children to play outdoors. They need to maximize the area using 180 feet of fencing on three sides of the yard. The quadratic equation $A=-2x^2+180x$gives the area, A, of the yard for the length, x, of the building that will border the yard. Find the length of the building that should border the yard to maximize the area, and then find the maximum area.

Short answer

The maximum area is 4050 square yards and the length of the building is 45 yards.

Step-by-step solution

Step 1. Given information.

The given equation is:

$A=-2x^2+180x$

Step 2. Find the maximum area.

$$\begin{gathered} y=a{x}^2+bx+c \\ A=-2x^2+180x \end{gathered}$$

The value of coefficient ais negative therefore we can say that the parabola opens down.

The quadratic function has a maximum.

The axis of symmetry is the line $x=-\frac{b}{2a}$.

Substitute the values of a, and binto the equation.

$$\begin{gathered} x=-\frac{180}{2\left(-2\right)} \\ =\frac{180}{4} \\ =45 \end{gathered}$$

Therefore, the maximum will occur when x is equal to 45..

The vertex is on the line of symmetry, so its x-coordinate will be $x=45$.

Now substitute the value of xinto the equation,

$$\begin{gathered} A\left(x\right)=-2x^2+180x \\ =-2{\left(45\right)}^2+180\left(45\right) \\ =-2\left(2025\right)+8100 \\ =-4050+8100 \\ =4050 \end{gathered}$$

It means that for 45 yards, the maximum area is 4050 square yards.