Q58. - Algebra 1

Question

A basketball is dropped from a height of 20 feet. It bounces to $\frac{\mathbf{1}}{\mathbf{2}}$ its height after each bounce. Draw a graph to represent the situation.

Step-by-Step Solution

Answer

The graph of the situation is:

Step 1. State the concept used.

In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. 

For example, the sequence $2, 6, 18, 54, ...$ is a geometric progression with common ratio 3.

The ration of $2^{nd}$ term and first term is called the common ratio.

Step 2. Find the next few terms of the series to get the series.

The ball loses $\frac{1}{2}$ its height after each bounce; hence it forms a geometric sequence. To determine a term (height after each bounce) multiply the previous term by $\frac{1}{2}$.

Initial height is the first term = 20 feet. 

$1^{st}$ term: $a_1=20$ and the common difference is $\frac{1}{2}$.

$2^{nd}$ term: $a_2=10$;

$3^{rd}$ term:  $a_3=5$

4^th term: $a_4=5\times \frac{1}{2}=2.5$;      

$5^{th}$ term: $a_5=2.5\times \frac{1}{2}=1.25$

$6^{th}$ term: $a_6=1.25\times \frac{1}{2}=0.625$ and so on.

Step 3. Plot the graph.

The GP series thus obtained is $20,10,5,2.5,1.25,\mathrm{0.625.....}$. Plot these points to obtained the graph of the situation: