A bouncing ball reaches a height of 54 inches at its first peak and 36 inches at its second peak. Which formula represents this scenario?

A. [tex]f(x) = 54\left(\frac{2}{3}\right)^x[/tex]
B. [tex]f(x) = 54\left(\frac{2}{3}\right)^{x-1}[/tex]
C. [tex]f(x) = \frac{2}{3}(54)^x[/tex]
D. [tex]f(x) = \frac{2}{3}(54)^{x-1}[/tex]



Answer :

To determine which formula represents the given scenario where a bouncing ball reaches a height of 54 inches at its first peak and 36 inches at its second peak, we need to understand the pattern of the ball's bounce.

### Step-by-Step Solution:

1. Identify the Height Pattern:
- The ball reaches 54 inches at its first peak.
- It then reaches 36 inches at its second peak.
- We can see that the heights follow a pattern of exponential decay.

2. Calculate the Ratio:
- The ratio of the second peak height to the first peak height is:
[tex]\[ \frac{36}{54} = \frac{2}{3} \][/tex]
- This ratio (decay factor) is consistent for each subsequent peak.

3. General Modeling of Height:
- The height of the ball at each peak can be modeled using a geometric sequence, where:
[tex]\[ H_n = H_1 \times \left( \frac{2}{3} \right)^{n-1} \][/tex]
- Here, [tex]\(H_1\)[/tex] is the height of the first peak, 54 inches, and [tex]\(n\)[/tex] is the peak number.

4. Identify the Formula:
- For peak [tex]\(n\)[/tex]:
[tex]\[ f(n) = 54 \left( \frac{2}{3} \right)^{n-1} \][/tex]

5. Match the Given Formulas:
- Compare the derived formula with the given options:
1. [tex]\(f(x) = 54 \left( \frac{2}{3} \right)^x\)[/tex]
2. [tex]\(f(x) = 54 \left( \frac{2}{3} \right)^{x-1}\)[/tex]
3. [tex]\(f(x) = \frac{2}{3} (54)^x\)[/tex]
4. [tex]\(f(x) = \frac{2}{3} (54)^{x-1}\)[/tex]

- The correct formula should be:
[tex]\[ 54 \left( \frac{2}{3} \right)^{x-1} \][/tex]
- This matches option 2.

### Conclusion:
The correct formula that represents the scenario is:
[tex]\[ f(x) = 54 \left( \frac{2}{3} \right)^{x-1} \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{f(x) = 54 \left( \frac{2}{3} \right)^{x-1}} \][/tex]