Let's analyze the problem step-by-step based on the functions provided and the goal of finding the correct cost expression.
1. Understanding the Functions:
- The first function provided is [tex]\( m(x) = \pi x^2 - 2\sqrt{2} x^2 \)[/tex]. This represents the area that requires mulch, where [tex]\( x \)[/tex] is the radius of the circular area. Essentially, [tex]\( m(x) \)[/tex] gives the area of the circle minus the area of the octagon inscribed within it.
- The second function, [tex]\( g(m) = 1.50 m \)[/tex], represents the cost of mulching the area requiring mulch. This indicates that the cost is \$1.50 per square foot of mulch.
2. Formulating the Cost Expression:
- Given the functions [tex]\( m(x) \)[/tex] and [tex]\( g(m) \)[/tex], we need to substitute [tex]\( m(x) \)[/tex] into [tex]\( g \)[/tex] to get the cost based on the radius of the circle:
[tex]\[
g(m(x)) = 1.50 \cdot m(x)
\][/tex]
- Substituting [tex]\( m(x) \)[/tex] into the equation:
[tex]\[
g(m(x)) = 1.50 \left( \pi x^2 - 2\sqrt{2} x^2 \right)
\][/tex]
3. Comparing with the Given Options:
- Now we compare the derived expression with the provided options:
1. [tex]\( 1.50 \left( \pi x^2 - 2 \sqrt{2} x^2 \right) \)[/tex]
2. [tex]\( \pi (1.50 x)^2 - 2 \sqrt{2} x^2 \)[/tex]
3. [tex]\( x (1.50 x)^2 - 2 \sqrt{2} (1.50 x)^2 \)[/tex]
4. [tex]\( 1.50 \left( \pi (1.50 x)^2 - 2 \sqrt{2} (1.50 x)^2 \right) \)[/tex]
Clearly, the derived equation matches the expression given in option 1.
Therefore, the correct 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]
