A homeowner has an octagonal gazebo inside a circular area. Each vertex of the gazebo lies on the circumference of the circular area. The area that is inside the circle, but outside the gazebo, requires mulch. This area is represented by the function [tex]$m(x)$[/tex], where [tex]$x$[/tex] is the length of the radius of the circle in feet. The homeowner estimates that he will pay [tex]$\$1.50$[/tex] per square foot of mulch. This cost is represented by the function [tex]$g(m)$[/tex], where [tex]$m$[/tex] is the area requiring mulch.
[tex]$\begin{array}{l}
m(x)=8 x^2-2 \sqrt{2} x^2 \\
g(m)=1.50 m
\end{array}$[/tex]
Which expression represents the cost of the mulch based on the radius of the circle?
A. [tex]$1.50\left(\pi x^2-2 \sqrt{2} x^2\right)$[/tex]
B. [tex]$\pi(1.50 x)^2-2 \sqrt{2} x^2$[/tex]
C. [tex]$1.50\left(\pi x^2-2 \sqrt{2} x^2\right)$[/tex]
D. [tex]$1.50\left(\pi(1.50 x)^2-2 \sqrt{2}(1.50 x)^2\right)$[/tex]
Answer (1)
Substitute the area function m ( x ) into the cost function g ( m ) .
The area function is m ( x ) = 8 x 2 − 2 × 2 x 2 .
The cost function is g ( m ) = 1.50 m .
The cost of the mulch based on the radius of the circle is 1.50 ( 8 x 2 − 2 2 x 2 ) .
Explanation
Analyze the problem and given data The problem provides the area requiring mulch as a function of the radius x of the circular area: m ( x ) = 8 x 2 − 2 2 x 2 . It also gives the cost function g ( m ) = 1.50 m , where m is the area requiring mulch. We need to find the expression that represents the cost of the mulch based on the radius of the circle.
Determine the approach To find the cost of the mulch based on the radius of the circle, we need to substitute the expression for m ( x ) into the cost function g ( m ) . This will give us the cost as a function of x , the radius of the circle.
Perform the substitution Substitute m ( x ) into g ( m ) : g ( m ( x )) = 1.50 × m ( x ) = 1.50 ( 8 x 2 − 2 2 x 2 )
State the final expression The expression representing the cost of the mulch based on the radius of the circle is 1.50 ( 8 x 2 − 2 2 x 2 ) .
Examples
Imagine you're designing a garden with a circular flower bed around a gazebo. Knowing the radius of the flower bed, you can use this formula to quickly calculate the cost of mulch needed to cover the area outside the gazebo. This helps in budgeting and planning your gardening project effectively, ensuring you don't overspend on materials. For example, if the radius is 5 feet, the cost would be approximately $1.50 * (8 * 5^2 - 2 * \sqrt{2} * 5^2) = $193.93.
