====== Question 15, Exercise 4.5 ======
Solutions of Question 15 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 15=====
To test its elasticity, a rubber ball is dropped into a $30 ft$ hollow tube that is calibrated so that the scientist can measure the height of each subsequent bounce. The scientist found that on each bounce, the ball rises to a height $\frac{2}{5}$ the height of the previous bounce. How far will the ball travel before it stops bouncing?

** Solution. **

Hight of ball $= 30 ft$ \\
First rebound $= 30 \times \frac{2}{5} = 12 ft$ \\ 
Second rebound $= 12 \times \frac{2}{5} = \frac{24}{5} ft$ \\
Third rebound $= \frac{24}{5} \times \frac{2}{5} = \frac{48}{25} ft$ \\

Let $D$ be the total distance covered by the ball. Then
$$D=30+2\left(12+\frac{24}{5}+\frac{24}{5}+... \right)$$
To find the sum of infinite geometric series 
$$
12+\frac{24}{5}+\frac{24}{5}+...
$$
We have $a_1=12$, $r=\frac{2}{5}$ with $|r|<1$, thus
\begin{align*}
S_\infty & = \frac{a_1}{1-r} \\
& = \frac{12}{1-\frac{2}{5}} = \frac{60}{3}
& = 20.
\end{align*}
Hence
$$D=30+2(20) = 70$$
Hence ball will travel $70\,\, ft$ before it stops bouncing. GOOD





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