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]
