Question

Some students performed an experiment in which they dropped a rubber ball from a height of 800 centimeters. They noticed that after each bounce, it reached 85% of its previous height. Which equation models the height, H, for n bounces? (1 point)Hn = 800(85)n − 1 where n = 1, 2, 3, ... Hn = 800(1.85)n − 1 where n = 1, 2, 3, ...Hn = 800(0.85)n − 1 where n = 1, 2, 3, ...Hn = 800(0.15)n − 1 where n = 1, 2, 3, ...

Answer 1

Let H represent the height.

Given that the initial height from which the rubber ball is dropped is 800 centimeters, this implies that

`H_1=800`

It is noticed that after each bounce, it reached 85% of its previous height. This implies that

`H_2=(85)/(100)* H_1=0.85H_1`

Similarly,

`H_3=(85)/(100)* H_2=0.85H_2`

Thus, since each height is attained by a common ratio, using the geometric sequence formula expressed as

`\begin{gathered} T_n=ar^(n-1) \\ where \\ T_n\implies H_n \\ a\implies H_1 \\ r\text{ is the common ratio between consecutive terms eevaluated as} \\ (H_2)/(H_1)=(0.85H_1)/(H_1)=0.85 \end{gathered}`

Thus, substituting these parameters into the geometric sequence formula, we have

`\begin{gathered} H_n=H_1(0.85)^(n-1) \\ where \\ H_1=800 \\ \Rightarrow H_n=800(0.85)^(n-1) \end{gathered}`

Hence, the equation model for the height H, for n bounces is

`\begin{gathered} \begin{equation*} H_n=800(0.85)^(n-1) \end{equation*} \\ where \\ n=1,2,3,... \end{gathered}`

The third option is the correct answer.