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Bouncing Ball

Problem Statement
A ball is dropped 520 feet from the roof of a building. Suppose that with each bounce, the ball goes up exactly x percent of its previous height. A man sitting at his desk on the second floor at a height of 15 feet above the ground, sees the ball pass him 17 times. Find x.

(Source: Mathematics Teaching in the Middle School, Feb 1999).

Problem setup

I am trying to find the percent of height that the ball goes up with each bounce.

Plans to Solve/Investigate the Problem

I must admit that I really did not think in terms of proportions, but in terms of exponents and logarithms.  But when I solved, I did have to divide, and percents are a form of a ratio.

Investigation/Exploration of the Problem
Initially,  the ball was dropped from a height of 520 feet.  With each drop the height of the ball decreased x%.  Cornell was sitting at his desk at work, and was RIDICULOUSLY off task! (Well he was if he counted the number of bounces - what a lame!)

Since I have just been teaching about exponents, I recognized that this could be done using exponents

The first drop            520ft

Second                      520*x ft

Third                         520*x*x ft = 520x2 ft

Fourth                        520*x*x*x ft = 520x3ft

Fifth                          520*x*x*x*x ft = 520x4ft
And so on

Seventeenth drop      520x16ft = 15

So I got that x≈80.13%

Extensions of the Problem

Discuss possible extensions for the problem and explore/investigate at least one of the extensions you discussed.

Author & Contact
Michelle Houston
mhouston@rockdale.k12.ga.us