SOLUTION: 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, circle but

Algebra ->  Rational-functions -> SOLUTION: 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, circle but       Log On


   

Question 1209185: 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, 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. The cost is represented by the function g(m), where m is the area requiring mulch.
m(x)=pi x^2
g(m)=1.50m
Write an expression that represents the cost of the mulch based on the radius of the circle.
Answer by ikleyn(53956) About Me  (Show Source):
You can put this solution on YOUR website!
.

An obvious trivial observation is that a regular octagon consists of 8 isosceles
congruent triangles with the common center. Each triangle has lateral sides of the length x,
equal to the radius of the circle, and the angle between two lateral sides is 

             360/8 = 45 degrees.


Therefore, the area of each separate triangle is  

    %281%2F2%29%2Ax%2Ax%2Asin%2845%5Eo%29 = %281%2F2%29x%5E2%2A%28sqrt%282%29%2F2%29 = x%5E2%2A%28sqrt%282%29%2F4%29.


Hence, the area of the octagon is 8 times this,  or  2x%5E2%2Asqrt%282%29.


Then the area of the circle outside the octagon is  

    pi%2Ax%5E2-2%2Asqrt%282%29%2Ax%5E2 = %28pi-2%2Asqrt%282%29%29%2Ax%5E2.


It is the expression you want to get.

Solved.