Question

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 \\(m(x)\\), where \\(x\\) is the length of the radius of the circle in feet. the homeowner estimates that he will pay \\$1.50 per square foot of mulch. this cost is represented by the function \\(g(m)\\), where \\(m\\) is the area requiring mulch.

\\m(x) = \pi x^2 - 2\sqrt{2}x^2\\
\\g(m) = 1.50m\\

which expression represents the cost of the mulch based on the radius of the circle?

• \\(1.50(\pi x^2 - 2\sqrt{2}x^2)\\)
• \\(\pi(1.50x)^2 - 2\sqrt{2}x^2\\)
• \\(\pi(1.50x)^2 - 2\sqrt{2}(1.50x)^2\\)
• \\(1.50(\pi(1.50x)^2 - 2\sqrt{2}(1.50x)^2)\\)

Explanation:

Identify the given functions

Using the Composite Functions knowledge point

$$
LATEXBLOCK0
$$

Form the composite function

Using the Composite Functions knowledge point

$$
LATEXBLOCK1
$$

Answer:

• (A) \(1.50(\pi x^2 - 2\sqrt{2}x^2)\) (Correct answer)
• (B) \(\pi(1.50x)^2 - 2\sqrt{2}x^2\)
• (C) \(\pi(1.50x)^2 - 2\sqrt{2}(1.50x)^2\)
• (D) \(1.50(\pi(1.50x)^2 - 2\sqrt{2}(1.50x)^2)\)