# Find Length of Rubber Band

• Dec 1st 2013, 07:40 AM
nycmath
Find Length of Rubber Band
If you wrap a rubber band around a ball 10 cm in diameter, then lift the band 1 cm off the ball all the way around, the band will be 6.28 cm longer. Now repeat the experiment around the earth (radius = 6.378 km), again lifting it up 1 cm all the way around. How much longer is the rubber band?
• Dec 1st 2013, 09:03 AM
sakonpure6
Re: Find Length of Rubber Band
Well, the circumference of the ball is C= (pi)d = 3.141... * 10 = about 31.41 cm. So when the rubber band is completely around the 10cm ball, it's length will be 31.41 cm. What I understood from the "lift the rubberband 1cm off the ball all the way around" is to increase the diameter.
If the new diameter is : 6.28 + 31.41 = pi (d)
37.69/ pi = d
d= 12.
There fore we can conclude that 2 cm were added to the diameter when we lift the rubber band 1cm away from the ball.
Hope this helped, now try doing the same thing using the radius of Earth.

If I am unclear on something, just tell me :)

Spoiler:
I think the solution is as follows: Find the ratio of the difference/normal circumference = 6.28/31.41 = 0.99, Find the circumference of the earth, then multiply it by the ratio to find an increase of the diameter by 8.011 km when the rubber band is removed 1 cm
• Dec 1st 2013, 09:36 AM
nycmath
Re: Find Length of Rubber Band
I will try later....
• Dec 1st 2013, 11:32 AM
Soroban
Re: Find Length of Rubber Band
Hello, nycmath!

This is a classic (very old) problem.

Quote:

If you wrap a rubber band around a ball 10 cm in diameter,
then lift the band 1 cm off the ball all the way around, the band will be 6.28 cm longer.
Now repeat the experiment around the Earth (radius = 6.378 km), again lifting it up 1 cm all the way around.
How much longer is the rubber band?

Let $\displaystyle d$ = the diameter of the ball.
The length of the rubber band is $\displaystyle \pi d$.

We lift the band $\displaystyle x$ cm off the ball.
We have a new circle with diameter $\displaystyle d + 2x$.
The length of the rubber band is $\displaystyle \pi(d+2x)$.

The difference is: .$\displaystyle \pi(d+2x) - \pi d \:=\:\pi d + 2\pi x - \pi d \:=\:2\pi x$

Since $\displaystyle x=1$ cm, the difference is: .$\displaystyle 2\pi\:\approx\:6.28$ cm.

Note that the diameter of the ball $\displaystyle (d)$ drops out.
Whether the ball is a BB or the Sun, the difference is always $\displaystyle 2\pi$ cm.

Surprise!
• Dec 1st 2013, 11:54 AM
nycmath
Re: Find Length of Rubber Band
Nicely done, Soroban.