Answered

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]$\$[/tex]1.50[tex]$ 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.

\[
\begin{array}{l}
m(x) = \pi x^2 - 2\sqrt{2} x^2 \\
g(m) = 1.50 m
\end{array}
\]

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. $[/tex]x(150 x)^2 - 2 \sqrt{2}(150 x)^2[tex]$

D. $[/tex]1.50\left(\pi(1.50 x)^2 - 2 \sqrt{2}(1.50 x)^2\right)$



Answer :

GinnyAnswer
To find the cost of the mulch based on the radius of the circle, we need to follow these steps:

1. Understand the function [tex]\( m(x) \)[/tex]:
[tex]\[ m(x) = \pi x^2 - 2\sqrt{2} x^2 \][/tex]
This function [tex]\( m(x) \)[/tex] represents the area that requires mulch, where [tex]\( x \)[/tex] is the radius of the circle.

2. Understand the function [tex]\( g(m) \)[/tex]:
[tex]\[ g(m) = 1.50 m \][/tex]
This function [tex]\( g(m) \)[/tex] represents the cost of the mulch per square foot, with a cost of $1.50 per square foot.

3. Substitute [tex]\( m(x) \)[/tex] into [tex]\( g(m) \)[/tex]:
We need to find [tex]\( g(m(x)) \)[/tex]:
[tex]\[ g(m(x)) = 1.50 (\pi x^2 - 2\sqrt{2} x^2) \][/tex]

So, by multiplying [tex]\( m(x) \)[/tex] by 1.50, we get:
[tex]\[ g(m(x)) = 1.50(\pi x^2 - 2\sqrt{2} x^2) \][/tex]

4. Simplify the Expression:
This result can be expanded to:
[tex]\[ g(m(x)) = 1.50\pi x^2 - 1.50 \cdot 2 \sqrt{2} x^2 \][/tex]
[tex]\[ g(m(x)) = 1.50\pi x^2 - 3\sqrt{2} x^2 \][/tex]

Thus, the expression that represents the cost of the mulch based on the radius of the circle is:
[tex]\[ 1.50\left(\pi x^2 - 2\sqrt{2} x^2\right) \][/tex]

Given the multiple-choice options, the correct choice is:
[tex]\[ \boxed{1.50\left(\pi x^2 - 2\sqrt{2} x^2\right)} \][/tex]

Other Questions