# Math Help - Geometric Series word problem - Bouncing Ball

1. ## Geometric Series word problem - Bouncing Ball

A ball is dropped from a height of 10 m. Each time it strikes the ground it bounces vertically to a height that is 3/4 of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often.

I cannot for the life of my figure this out.

I tried the obvious way: making the geometric series $(10)* (\frac{3}{4})^k$ and then found the convergence as k --> infinity, and got 40.

Next, because the ball bounces twice, I thought perhaps I needed a 2. I made the following geometric series,
$(2)(10)*\frac{3}{4}$, used the convergence formula, and got 80.

I believe my second attempt makes more sense, but it doesn't equal 70 m.

So, what am I doing wrong?

Thanks!

2. You have got it – except that you don’t multiply the first distance (the initial drop of $10\ \mathrm m)$ by $2.$ The distance is therefore

$10+2(10)\left(\frac34\right)+2(10)\left(\frac34\ri ght)^2+2(10)\left(\frac34\right)^3+\cdots$

3. Originally Posted by Truthbetold
A ball is dropped from a height of 10 m. Each time it strikes the ground it bounces vertically to a height that is 3/4 of the preceding height. Find the total distance the ball will travel if it is assumed to bounce infinitely often.

I cannot for the life of my figure this out.

I tried the obvious way: making the geometric series $(10)* (\frac{3}{4})^k$ and then found the convergence as k --> infinity, and got 40.

Next, because the ball bounces twice, I thought perhaps I needed a 2. I made the following geometric series,
$(2)(10)*\frac{3}{4}$, used the convergence formula, and got 80.

I believe my second attempt makes more sense, but it doesn't equal 70 m.

So, what am I doing wrong?

Thanks!
HI

You have the right idea .

10 + 2(7.5)+2(5.625)+...

=10+2[7.5+5.625+...]

=10+2[7.5/(1-0.75)]

=70