Circle Area and Circumference

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

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