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.

Answer 1

To find the area left to mow after a certain number of hours of work, substitute the value of x (the number of hours) into the given equation ŷ = 1350 - 1.2x. After 20 hours, there are 1326 acres left to mow. After 100 hours, there are 1230 acres left to mow. It takes 1125 hours to mow all the lawns.

Step-by-step explanation:

Answer:

To find the area left to mow after a certain number of hours of work, we can substitute the value of x (the number of hours) into the given equation ŷ = 1350 - 1.2x.

31. After 20 hours of work, we substitute x = 20 in the equation ŷ = 1350 - 1.2x:
ŷ = 1350 - 1.2(20) = 1350 - 24 = 1326 acres
So, there are 1326 acres left to mow after 20 hours of work.
32. After 100 hours of work, we substitute x = 100 in the equation ŷ = 1350 - 1.2x:
ŷ = 1350 - 1.2(100) = 1350 - 120 = 1230 acres
So, there are 1230 acres left to mow after 100 hours of work.
33. To find the number of hours it takes to mow all the lawns, we set ŷ = 0 and solve for x:
0 = 1350 - 1.2x
1.2x = 1350
x = 1350 / 1.2
x = 1125
So, it takes 1125 hours to mow all the lawns.

Answer 2

The equation of cost of mulch is
`1.50* (\pi x^(2)-2√(2)x^(2))\\\\`.

Step-by-step explanation:

The equations m (x) and g (m) are as follows:

`m(x)=\pi x^(2)-2√(2)x^(2)\\g(m)=1.50m`

It is provided that the cost of mulch requires is represented by the function g(m), where m is the area requiring mulch.

To compute the equation of cost of mulch based on the radius of the circle substitute the value of m (x) in g (m).

The equation of cost of mulch is:

`g(m) = 1.50m`

`=1.50* m(x)\\\\=1.50* (\pi x^(2)-2√(2)x^(2))\\\\`

Thus, the equation of cost of mulch is
`1.50* (\pi x^(2)-2√(2)x^(2))\\\\`.