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

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

Sagot :

GinnyAnsweridn
To determine the cost of mulch based on the radius [tex]\(x\)[/tex] of the circle, let's follow the given functions step by step:

1. Calculate the area requiring mulch, [tex]\(m(x)\)[/tex]:

The function [tex]\(m(x)\)[/tex] gives the area needing mulch:
[tex]\[ m(x) = \pi x^2 - 2 \sqrt{2} x^2 \][/tex]

2. Calculate the cost based on the area, [tex]\(g(m)\)[/tex]:

The cost function [tex]\(g(m)\)[/tex] is given as:
[tex]\[ g(m) = 1.50 m \][/tex]

3. Combine the functions:

The cost of the mulch, based on the radius [tex]\(x\)[/tex] of the circle, can be represented by substituting [tex]\(m(x)\)[/tex] into [tex]\(g(m)\)[/tex]:
[tex]\[ g(m(x)) = 1.50 \left(\pi x^2 - 2 \sqrt{2} x^2\right) \][/tex]

Therefore, the expression representing 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]

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