Question

topic 6-1 homework name hour 4
1.) you drop a ball from a height of 10 feet. each bounce the ball gains only 90% of its height back. complete the table and graph below.

is this exponential growth or decay?
how high will the ball bounce after 10 bounces?
after 20 bounces?

Explanation:

Step1: Define exponential decay formula

The height after $n$ bounces follows $h(n) = h_0 \times r^n$, where $h_0=10$ ft, $r=0.9$.

Step2: Calculate height for bounce 1

Multiply initial height by 0.9.
$h(1) = 10 \times 0.9 = 9$

Step3: Calculate height for bounce 2

Multiply bounce 1 height by 0.9.
$h(2) = 9 \times 0.9 = 8.1$

Step4: Calculate height for bounce 3

Multiply bounce 2 height by 0.9.
$h(3) = 8.1 \times 0.9 = 7.29$

Step5: Calculate height for bounce 4

Multiply bounce 3 height by 0.9.
$h(4) = 7.29 \times 0.9 = 6.561$

Step6: Calculate height for bounce 5

Multiply bounce 4 height by 0.9.
$h(5) = 6.561 \times 0.9 = 5.9049$

Step7: Calculate height for bounce 6

Multiply bounce 5 height by 0.9.
$h(6) = 5.9049 \times 0.9 = 5.31441$

Step8: Classify growth/decay

Since $r<1$, it is exponential decay.

Step9: Calculate height after 10 bounces

Use the exponential formula for $n=10$.
$h(10) = 10 \times 0.9^{10} \approx 3.487$

Step10: Calculate height after 20 bounces

Use the exponential formula for $n=20$.
$h(20) = 10 \times 0.9^{20} \approx 1.216$

Answer:

Completed Table:

Bounce	Height (ft)
1	9
2	8.1
3	7.29
4	6.561
5	5.9049
6	5.31441

Additional Answers:

• This is exponential decay.
• Height after 10 bounces: $\approx 3.49$ feet
• Height after 20 bounces: $\approx 1.22$ feet