,
and 
are relatively prime positive integers. Find 
.
Y by centslordm
A roll of tape is 
inches in diameter and is wrapped around a ring that is 
inches in diameter. A cross section of the tape is shown in the figure below. The tape is 
inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 
inches.
![[asy]
/* AMC8 P22 2024, revised by Teacher David */
size(120);
pair o = (0,0);
real r1 = 1;
real r2 = 2;
filldraw(circle(o, r2), mediumgray, linewidth(1pt));
filldraw(circle(o, r1), white, linewidth(1pt));
draw((-2,-2.6)--(-2,-2.4));
draw((2,-2.6)--(2,-2.4));
draw((-2,-2.5)--(2,-2.5), L=Label("4 in."));
draw((-1,0)--(1,0), L=Label("2 in.", align=(0,1)), arrow=Arrows());
draw((2,0)--(2,-1.3), linewidth(1pt));
[/asy]](//latex.artofproblemsolving.com/e/d/b/edbca8b1b4e34ef5fac9cb2ad93cbde53f05a803.png)
inches in diameter and is wrapped around a ring that is 
inches in diameter. A cross section of the tape is shown in the figure below. The tape is 
inches thick. If the tape is completely unrolled, approximately how long would it be? Round your answer to the nearest 
inches.![[asy]
/* AMC8 P22 2024, revised by Teacher David */
size(120);
pair o = (0,0);
real r1 = 1;
real r2 = 2;
filldraw(circle(o, r2), mediumgray, linewidth(1pt));
filldraw(circle(o, r1), white, linewidth(1pt));
draw((-2,-2.6)--(-2,-2.4));
draw((2,-2.6)--(2,-2.4));
draw((-2,-2.5)--(2,-2.5), L=Label("4 in."));
draw((-1,0)--(1,0), L=Label("2 in.", align=(0,1)), arrow=Arrows());
draw((2,0)--(2,-1.3), linewidth(1pt));
[/asy]](http://latex.artofproblemsolving.com/e/d/b/edbca8b1b4e34ef5fac9cb2ad93cbde53f05a803.png)
This post has been edited 2 times. Last edited by pog, Oct 20, 2024, 8:33 PM
revolutions. So the average length of tape is 
Multiplying gives 
This is closest to 
,
,
![$\frac{\pi\left[{2^2} - {1^2}\right]}{dr} = \frac{3\pi}{0.015}= \frac{3 \dot 1000 \dot \pi}{15} = \frac{3 \dot 200 \dot \pi}{3} = 200\pi \approx 628 \boxed{B}$](http://latex.artofproblemsolving.com/c/f/5/cf54db1eee7a89aea2312919fbbc8ed6901cb3dc.png)
revolutions. The average length of tape is 
This is closest to B 
