A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by 10 pi feet, how wide is the track in feet?

Thanks

AnonymousConfusedGuy
Apr 11, 2018

#1**+1 **

Note that if the circumference of the larger circle is 10pi feet more....the track must be 5 feet wide

Proof....let the radius of the smaller circle = r

Then....let the radius of the larger circle be r + 5

So.... circumference of the smaller circle is 2pi * r

And the circumference of the larger circle is 2pi ( r + 5) = 2pi * r + 10pi

Subtracting the smaller circumference from the larger, we get

[ 2 pi * r + 10 pi ] - [ 2 pi * r ] = 10pi

CPhill
Apr 11, 2018

#2**+1 **

Here's a more formal way to do this, ACG

Let the radius of the smaller circle = R

Then the radius of the larger circle is just the radius of the smaller circle plus the width of the track, W

So

The circumference of the smaller circle plus 10pi ft = the circumference of the larger circle

So we have

2pi(R) + 10pi = 2pi (R + W)

2pi (R) + 10 pi = 2pi (R) + 2pi(W) subtract 2pi (R) from both sides

10 pi = 2pi (W)

10 = 2W

5 (feet) = W = width of the track

CPhill
Apr 12, 2018