Question

One day during practice, LeBron decided to collect some data on the height of a basketballs bounce. He first dropped the basketball from an initial height of 20 feet. At its first bounce, he recorded that the ball reached a height of 10 feet. At its second bounce, he recorded a height of 5 feet, and so on. His data is displayed in the table below.
1. If the ball were dropped from a different initial height, would the common ratio be different? Explain your reasoning.

Answer 1

2. What is the height of the ball on the fifth bounce? Use the geometric sequence formula, an = a1rn – 1 and show your work.

3. What is the total distance of the height the ball has traveled during the first five heights shown in the chart?
Use the geometric series formula, Sn = (a1 – a1rn) / (1 – r)
and show your work.

Answer 2

im so confused

Answer 3

\[
a_n = a_1r^n
\]

Answer 4

First of all, \(r\) is just the ratio between the heights of any two bounces. For geometric series this will always be the same.

Answer 5

ok

Answer 6

So, in general \[
r = \frac{a_{n}}{a_{n-1}}
\]And to be specific: \[
r = \frac{a_2}{a_1} = \frac{a_3}{a_2} = \frac{a_4}{a_3}
\]

Answer 7

so how do i fit that in with height of the ball

Answer 8

The term \(a_n\) just represents the height before the \(n\)th bounce. For example \(a_1\) is the height before the first bounce. Since the first height is 20, we say \(a_1=20\)

Answer 9

i got Starting height 20 feet
First bounce 10 feet
Second bounce 5 feet
Third bounce 2.5 feet
Fourth bounce 1.25 feet

Answer 10

Before the second bounce (after the first), they say the height is 10, so we can say \(a_2 = 10\)

Answer 11

right?

Answer 12

what im trying to find are the 3 questions if balls dropping effect the ratio an such

Answer 13

Well, you have determined the pattern correctly implicitly at least.

Answer 14

ok erm so if the ball were dropped from a different initial height, would the common ratio be different?

Answer 15

.....?

Answer 16

no, the ratio wouldn't be any different