Find the cost of resurfacing the track.

Question

An elliptical shaped track is shown. The cost of resurfacing the track is $0.45 per square foot. Find the total cost of resurfacing the track. Round the answer to the nearest ten dollars.

 

Track thickness= 10'-3''

Track height= 165'-0''

Track width= 210'-0''

Answer 1

So, I'll assume that this problem means that the track is essentially an ellipse with an ellipse-shaped hole in the middle of it. That means, to figure out the surface area of the track, we can calculate the surface area of the larger ellipse and subtract the area of the smaller ellipse (the hole) from it.

 

 

First, let's figure out the surface area of the larger ellipse. The area of an ellipse is:

 

A = πab,

 

where a is half the length and b is half the width.

 

For your problem, we'll use height and width instead of length and width. So a = 1/2(165) = 82.5 feet and b = 1/2(210) = 105 feet. Plugging those in, you get:

 

A_L = π(82.5)(105)

A_L = 8662.5π

 

The larger ellipse's surface area is 8,662.5π square feet.

 

 

The smaller ellipse is 20 feet 6 inches smaller in height and width. We can just use a and b from before (each of which is half the height or width) and subtract the track's thickness from each once. The track's thickness is 10 feet 3 inches = 10 and 3/12 feet = 10.25 feet. That means our new a and b are a = 82.5 - 10.25 = 72.25 feet and b = 105 - 10.25 = 94.75 feet. Plugging those in, you get:

 

A_S = π(72.25)(94.75)

A_S = 6845.7π

 

The smaller ellipse's surface area is 6,845.7π square feet.

 

 

In total, you want to resurface:

 

A_L - A_S = 8,662.5π - 6,845.7π = 1,816.8π square feet.

 

 

This is approximately 5,707.6 square feet of track. If each square foot costs $0.45, the total cost will be:

 

5707.6(0.45) = 2,568.42

 

Rounded to the nearest ten dollars, that's a total of $2,570.