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.
mc027-1.jpg
mc027-2.jpg
Which expression represents the cost of the mulch based on the radius of the circle?
mc027-3.jpg
mc027-4.jpg
mc027-5.jpg
mc027-6.jpg

Answer 1

Area of circle = πx^2.
Area of a regular octagon with circumradius x = (1/2)·8·x^2·sin(2π/8) = 4x^2·sin(π/4) = 2√2 x^2.
Mulch area m(x) = πx^2 − 2√2 x^2.
Cost = 1.50·m(x) = 1.5(πx^2 − 2√2 x^2) = (3/2)(π − 2√2)x^2 dollars.