To determine the formula that represents the described scenario of a bouncing ball, let's analyze the heights provided and identify the pattern they follow step-by-step.
### Step-by-Step Solution:
1. Initial Heights Given:
- First peak: [tex]\( h_1 = 54 \)[/tex] inches
- Second peak: [tex]\( h_2 = 36 \)[/tex] inches
- Third peak: [tex]\( h_3 = 24 \)[/tex] inches
2. Identify the pattern:
- We observe that the heights decrease in a specific way. Let's calculate the common ratio [tex]\( r \)[/tex] between consecutive heights.
- Common ratio [tex]\( r \)[/tex] between the first and second peak:
[tex]\[
r = \frac{h_2}{h_1} = \frac{36}{54} \approx 0.6667 \approx \frac{2}{3}
\][/tex]
- Common ratio [tex]\( r \)[/tex] between the second and third peak:
[tex]\[
r = \frac{h_3}{h_2} = \frac{24}{36} \approx 0.6667 \approx \frac{2}{3}
\][/tex]
3. General Formula Using the Pattern:
- The pattern shows that each height is multiplied by the common ratio [tex]\( \frac{2}{3} \)[/tex] to get the next height.
- As such, we are dealing with a geometric sequence where the initial height [tex]\( h_0 \)[/tex] (first term) is 54 inches and the common ratio [tex]\( r \)[/tex] is [tex]\( \frac{2}{3} \)[/tex].
4. Constructing the Formula:
- For a geometric sequence, the [tex]\( n \)[/tex]-th term [tex]\( a_n \)[/tex] can be defined as:
[tex]\[
a_n = a_0 \cdot r^n
\][/tex]
- In our case, the height at peak [tex]\( x \)[/tex] can be expressed as:
[tex]\[
f(x) = 54 \left(\frac{2}{3}\right)^x
\][/tex]
where [tex]\( x \)[/tex] is the number of peaks after the first peak.
5. Conclusion:
- Therefore, the formula that correctly represents the height of the ball at each peak follows the exponential decay pattern as described and is given by:
[tex]\[
\boxed{f(x) = 54 \left(\frac{2}{3}\right)^x}
\][/tex]
Thus, the formula [tex]\( f(x) = 54 \left(\frac{2}{3}\right)^x \)[/tex] accurately models the scenario of the bouncing ball and matches the given decreasing sequence of heights.
