To solve the problem, let's break it down step-by-step.
1. Finding the Area Inside but Outside the Gazebo (m(x)):
The given expression for the area that requires mulch is [tex]\( m(x) = x^2 - 2\sqrt{2} x^2 \)[/tex].
We can simplify this as follows:
[tex]\[
m(x) = x^2(1 - 2\sqrt{2})
\][/tex]
2. Finding the Total Cost of the Mulch (g(m)):
The cost function provided is [tex]\( g(m) = 1.50 \cdot m \)[/tex].
We need to substitute [tex]\( m(x) \)[/tex] into [tex]\( g(m) \)[/tex]:
[tex]\[
g(m(x)) = 1.50 \cdot (x^2(1 - 2\sqrt{2}))
\][/tex]
Let's break it down:
[tex]\[
g(m(x)) = 1.50 \cdot x^2 \cdot (1 - 2\sqrt{2})
\][/tex]
Therefore, the expression that represents the cost of the mulch based on the radius of the circle [tex]\( x \)[/tex] is:
[tex]\[
1.50 \left( x^2 - 2\sqrt{2} x^2 \right)
\][/tex]
Among the given options, the correct one that matches our derived expression is:
[tex]\[
\boxed{1.50\left(x x^2 - 2\sqrt{2} x^2\right)}
\][/tex]
