Henry Ernest Dudeney/Puzzles and Curious Problems/223 - The Tower of Pisa/Solution

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Puzzles and Curious Problems by Henry Ernest Dudeney: $223$

The Tower of Pisa
Suppose you were on the top of the Tower of Pisa, at a point where it leans exactly $179$ feet above the ground.
Suppose you were to drop an elastic ball from there such that on each rebound it rose exactly one-tenth of the height from which it fell.
What distance would the ball travel before it came to rest?


Solution

$218$ feet $9 \tfrac 1 3$ inches.


Proof

Let $d$ feet be the distance travelled.

First the ball falls $179$ feet.

After the first bounce, it goes up again $\dfrac {179} {10}$ feet and down again the same distance.

Similarly, after the second bounce, it goes up again $\dfrac {179} {10^2}$ feet and down again the same distance.

After the $n$th bounce, it goes up again $\dfrac {179} {10^n}$ feet and down again the same distance.

So the total distance travelled is given by:

$\ds d = 179 + 2 \times \dfrac {179} {10} \sum_{n \mathop \ge 0} \paren {\dfrac 1 {10} }^n$

which from Sum of Infinite Geometric Sequence gives:

\(\ds d\) \(=\) \(\ds 179 + 2 \times \dfrac {179} {10} \sum_{n \mathop \ge 0} \paren {\dfrac 1 {10} }^n\)
\(\ds \) \(=\) \(\ds 179 + 2 \times \dfrac {179} {10} \dfrac 1 {1 - 1/10}\) Sum of Infinite Geometric Sequence
\(\ds \) \(=\) \(\ds 179 + 2 \times \dfrac {179} {10} \dfrac {10} 9\) simplification
\(\ds \) \(=\) \(\ds 179 + 39 \tfrac 7 9\) more simplification
\(\ds \) \(=\) \(\ds 218 \tfrac 7 9\) more simplification

The answer given follows after we recall that $1$ foot is $12$ inches.

$\blacksquare$


Sources

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