Question

a ball is dropped from a height of 14 ft and bounces 80% of its previous height on each bounce. how high off the ground is the ball at the top of the 6th bounce?
the ball will bounce \square ft on the sixth bounce.
(round to one decimal place as needed.)

Explanation:

Step1: Identify the formula

This is a geometric sequence problem where the first term \( a_1 = 14 \) (initial height) and the common ratio \( r = 0.8 \) (80% of previous height). The formula for the \( n \)-th term of a geometric sequence is \( a_n=a_1\times r^{(n - 1)} \).

Step2: Substitute values for the 6th bounce

For the 6th bounce, \( n = 6 \). So we substitute \( a_1 = 14 \), \( r=0.8 \) and \( n = 6 \) into the formula:
\( a_6=14\times(0.8)^{(6 - 1)}=14\times(0.8)^{5} \)

Step3: Calculate \( (0.8)^5 \)

First, calculate \( 0.8^5=0.8\times0.8\times0.8\times0.8\times0.8 = 0.32768 \)

Step4: Multiply by 14

Then, \( 14\times0.32768 = 4.58752 \)

Step5: Round to one decimal place

Rounding \( 4.58752 \) to one decimal place gives \( 4.6 \)

Answer:

\( 4.6 \)