Question

a ball is dropped from a height of 25 feet and allowed to continue to bounce until it eventually comes to a rest. each time the ball bounces it rebounds to $\frac{4}{5}$ of its previous height. after which bounce is the balls rebound height 10.24 ft? bounce 3 bounce 5 bounce 4 bounce 6

Explanation:

Step1: Identify the geometric - sequence formula

The height of the ball after each bounce forms a geometric sequence with the formula $a_n=a_1r^{n - 1}$, where $a_1$ is the initial height after the first - bounce, $r$ is the common ratio, and $n$ is the number of the bounce. The ball is dropped from 25 feet, and after the first bounce, $a_1 = 25\times\frac{4}{5}$, and $r=\frac{4}{5}$. So the formula for the height $h_n$ after the $n$th bounce is $h_n = 25\times(\frac{4}{5})^n$.

Step2: Set up the equation

We want to find $n$ when $h_n=10.24$. So we set up the equation $25\times(\frac{4}{5})^n = 10.24$.
First, divide both sides of the equation by 25: $(\frac{4}{5})^n=\frac{10.24}{25}=0.4096$.

Step3: Rewrite the numbers

We know that $\frac{4}{5}=0.8$ and $0.4096 = 0.8^4$. So the equation $(\frac{4}{5})^n=0.4096$ can be rewritten as $(0.8)^n=(0.8)^4$.

Step4: Solve for $n$

Since the bases are the same ($0.8$), then $n = 4$.

Answer:

Bounce 4