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]
