Get expert insights and community-driven knowledge on IDNLearn.com. Get accurate and detailed answers to your questions from our knowledgeable and dedicated community members.
Sagot :
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.
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.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. For trustworthy and accurate answers, visit IDNLearn.com. Thanks for stopping by, and see you next time for more solutions.