Certainly! Let's go through this problem step-by-step to determine the total vertical distance the rubber ball has traveled after completing its fifth bounce.
Step 1: Analyze the initial drop and first bounce.
- The ball is initially dropped from a height of 10 feet, so the first downward distance is:
[tex]\[
\text{Initial drop} = 10 \text{ feet}
\][/tex]
Step 2: Calculate the height after each bounce.
- Each subsequent bounce reaches [tex]\(\frac{4}{5}\)[/tex] of the height of the previous bounce.
Step 3: Compute the total distance traveled for the first drop and the first bounce.
- After the initial drop, the ball bounces back up to:
[tex]\[
\text{Height after first bounce} = 10 \times \frac{4}{5} = 8 \text{ feet}
\][/tex]
- The ball then falls back down the same distance (8 feet). So, for the first bounce, the ball travels:
[tex]\[
\text{Distance for the first bounce} = 8 \text{ feet (up)} + 8 \text{ feet (down)} = 16 \text{ feet}
\][/tex]
Step 4: Sum the distances for each subsequent bounce.
- We perform this for the second to fifth bounces, each time multiplying the height by [tex]\(\frac{4}{5}\)[/tex].
Step 5: Calculate the distance for each bounce height.
- Second bounce:
[tex]\[
\text{Height} = 8 \times \frac{4}{5} = 6.4 \text{ feet}
\][/tex]
[tex]\[
\text{Distance for second bounce} = 6.4 \text{ feet (up)} + 6.4 \text{ feet (down)} = 12.8 \text{ feet}
\][/tex]
- Third bounce:
[tex]\[
\text{Height} = 6.4 \times \frac{4}{5} = 5.12 \text{ feet}
\][/tex]
[tex]\[
\text{Distance for third bounce} = 5.12 \text{ feet (up)} + 5.12 \text{ feet (down)} = 10.24 \text{ feet}
\][/tex]
- Fourth bounce:
[tex]\[
\text{Height} = 5.12 \times \frac{4}{5} = 4.096 \text{ feet}
\][/tex]
[tex]\[
\text{Distance for fourth bounce} = 4.096 \text{ feet (up)} + 4.096 \text{ feet (down)} = 8.192 \text{ feet}
\][/tex]
- Fifth bounce:
[tex]\[
\text{Height} = 4.096 \times \frac{4}{5} = 3.2768 \text{ feet}
\][/tex]
[tex]\[
\text{Distance for fifth bounce} = 3.2768 \text{ feet (up)} + 3.2768 \text{ feet (down)} = 6.5536 \text{ feet}
\][/tex]
Step 6: Add all the distances together.
- Total distance = initial drop distance + distance for 1st bounce + distance for 2nd bounce + distance for 3rd bounce + distance for 4th bounce + distance for 5th bounce
[tex]\[
\text{Total distance} = 10 + 16 + 12.8 + 10.24 + 8.192 + 6.5536 = 63.7856 \text{ feet}
\][/tex]
So, the total vertical distance the ball has traveled by the time it hits the surface for its fifth bounce is [tex]\(63.7856\)[/tex] feet.
