To solve this problem, we need to understand how to model the height of each successive bounce of the basketball. The key pieces of information are:
1. The basketball is initially dropped from a height of 120 feet.
2. Each bounce is 20% shorter than the previous one, which means each bounce retains 80% of the height of the previous bounce.
Given this, we need an explicit formula that describes the height of the basketball after each bounce [tex]\( n \)[/tex].
### Step-by-Step Solution:
1. Initial Height:
The initial height from which the basketball is dropped is 120 feet. This is the height before the first bounce.
2. Common Ratio:
Since each bounce is 20% shorter than the previous one, the ball retains 80% of its height after each bounce. This retention ratio can be expressed as a decimal: [tex]\( 0.80 \)[/tex].
3. General Formula for Geometric Sequence:
The height of the basketball after [tex]\( n \)[/tex] bounces forms a geometric sequence because each term is a constant multiple of the previous term. The general formula for the [tex]\( n \)[/tex]-th term of a geometric sequence is:
[tex]\[
a_n = a \cdot r^{n-1}
\][/tex]
where:
- [tex]\( a \)[/tex] is the initial term (120 feet in this case),
- [tex]\( r \)[/tex] is the common ratio (0.80 in this case), and
- [tex]\( n \)[/tex] is the term number (the number of bounces).
4. Substitute the Values:
Plugging in our values into the formula, we get:
[tex]\[
a_n = 120 \cdot (0.80)^{n-1}
\][/tex]
5. Answer:
Therefore, the explicit formula that describes the height of each bounce is:
[tex]\[
a_n = 120(0.80)^{n-1}
\][/tex]
Given the multiple-choice options, the correct one is:
[tex]\[
a_n = 120(0.80)^{n-1}
\][/tex]
So, the correct answer is:
[tex]\[
a_n=120(0.80)^{n-1}
\][/tex]
