Solution to Ball Bounce Problem Quiz #S2-05
REMEMBER! There are two ways to think about this problem: the exponential function approach and
the geometric series approach.
We’ll draw the situation first…
Each bounce distance has to be multiplied by 2 because the ball goes up the distance then back down
the same distance.
The Exponential Function Approach…
We could model this by using an exponential function 𝑦 = 12 ∗ (0.12)𝑥 where y is the height of the
ball and x is the bounce where that height occurs. To make this easier, let’s use a table!
x (bounce #)
y (height of the ball)
0
1
2
3
4
5
6
12
1.44
0.1728
0.020736
0.00248832
0.0002985984
0.000035831808
From here, we just add all of the heights from bounce 0 to bounce #6. REMEMBER to multiply bounces
#1-5 by 2 because it went up the distance and down the same distance.
Distance = 12 + 2 ( 1.44 + 0.1728 + 0.020736 + 0.00248832 + 0.0002985984 + 0.000035831808)
Distance = 15.27 feet  FINAL ANSWER!
The Geometric Series Approach…
Observing the picture above, we see that part of this problem is a geometric sequence. The term
number “n” is the bounce number.
We’ll use the formula from the formula chart…
𝑛
∑ 𝑎𝑘 =
𝑘=1
𝑎1 ∗ (1 − 𝑟 𝑛 )
1−𝑟
For us, a1 is 1.44, the common ratio is 0.12, and n = 6 because we want to add the first 6 bounces.
6
∑ 1.44 ∗ (0.12)𝑛 =
𝑛=1
1.44 ∗ (1 − 0.126 )
= 1.63636
1 − 0.12
REMEMBER! We need to multiply the sum of these bounces by 2…
1.63636 * 2 = 3.27272
Finally, add the initial 12…
12 + 3.27272 = 15.27 feet  FINAL ANSWER!
Two different methods lead to the same answer. Take your pick!