a bouncing ball reaches a height of 27 feet at its first peak, 18 feet at its second peak, and 12 feet at…

Explanation:

Step1: Identify the sequence type

The heights are 27, 18, 12. Calculate the ratio between consecutive terms. $\frac{18}{27}=\frac{2}{3}$ and $\frac{12}{18}=\frac{2}{3}$. It's a geometric - sequence with a common ratio $r = \frac{2}{3}$.

Step2: Use the geometric - sequence formula

The formula for the $n$th term of a geometric sequence is $a_n=a_1r^{n - 1}$, where $a_1$ is the first term, $r$ is the common ratio, and $n$ is the term number. Here, $a_1 = 27$, $r=\frac{2}{3}$, and we want to find the 4th term ($n = 4$).

Step3: Calculate the fourth - term

$a_4=a_1r^{4 - 1}=27\times(\frac{2}{3})^{3}=27\times\frac{8}{27}=8$.

Answer:

The height of the ball at its fourth peak is 8 feet.