Question

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 m(x), where x is the length of the radius of the circle in feet. the homeowner estimates that he will pay $1.50 per square foot of mulch. this cost is represented by the function g(m), where m is the area requiring mulch.

m(x) = (pi x^{2}-2sqrt{2}x^{2})
g(m) = 1.50m

which expression represents the cost of the mulch based on the radius of the circle?

1.50((pi x^{2}-2sqrt{2}x^{2}))
(pi(1.50x)^{2}-2sqrt{2}x^{2})
(pi(1.50x)^{2}-2sqrt{2}(1.50x)^{2})
1.50((pi(1.50x)^{2}-2sqrt{2}(1.50x)^{2}))

Explanation:

Step1: Identify the functions

We have the area function $m(x)=\pi x^{2}-2\sqrt{2}x^{2}$ (area requiring mulch in terms of radius $x$) and the cost - per - area function $g(m) = 1.50m$ (cost in terms of area $m$).

Step2: Find the cost function in terms of $x$

To find the cost of the mulch based on the radius $x$ of the circle, we need to substitute $m = m(x)$ into $g(m)$. So we replace $m$ in $g(m)$ with $\pi x^{2}-2\sqrt{2}x^{2}$. Then $g(m(x))=1.50(\pi x^{2}-2\sqrt{2}x^{2})$.

Answer:

$1.50(\pi x^{2}-2\sqrt{2}x^{2})$