A ball is dropped from a height of 16 feet. Each time the ball dropped h feet, it rebounds 0.81h feet. Find the total vertical distanced travelled by the ball.
Sorry. I see that r = 0.81 according to the problem, not 0.8.
Well substituting again,
d = 16 + (0.81)(32 / (1 - 0.81))
d = 152.4 ft.
This is an example of infinite geometric sequence.
Formula for sum of infinite geom sequence:
S = a1 / (1 - r)
Total distance traveled:
d = (distance ball dropped) + (1st bounce up) + (1st bounce down) + (2nd bounce up) + (2nd bounce down) + ...
d = 16 + 16(0.8) + 16(0.8) + 16(0.8)(0.8) + 16(0.8)(0.8) + ...
d = 16 + 32(0.8) + 32(0.8)(0.8) + ...
d = 16 + (0.8)(32 + 32(0.8) + 32(0.8)(0.8) + ...)
we can see here that the infinite geometric sequence applies on the terms after 16, specifically on the (32 + 32(0.8) + 32(0.8)(0.8) + ...). Thus, a1 = 32 and r = 0.8. Using the formula,
d = 16 + (0.8)(32 / (1 - 0.8))
d = 144 ft.
hope this helps? `u`
To find the total vertical distance traveled by the ball, we need to determine how many times the ball rebounds and calculate the distance traveled during each rebound. Let's break down the steps:
1. Start by dropping the ball from a height of 16 feet.
2. The ball will travel 16 feet downward during the initial drop.
3. After reaching the ground, the ball will rebound. We need to determine the height from which it rebounds.
4. Given that the ball rebounds 0.81 times the height from which it was dropped, we find that it rebounds 0.81 * 16 = 12.96 feet.
5. The ball will now travel 12.96 feet upward in the first rebound.
6. After reaching the highest point of the first rebound, the ball will fall back downward. It will travel 12.96 feet downward again.
7. Upon reaching the ground again, the ball will rebound for a second time. It will rebound 0.81 times the height it reached on the first rebound.
8. The height for the second rebound is 0.81 * 12.96 = 10.5136 feet
9. The ball will travel 10.5136 feet upward during the second rebound.
10. After reaching the peak of the second rebound, the ball will fall back downward. It will travel 10.5136 feet downward again.
11. The process of rebounding and falling will continue until the height becomes negligible.
To find the total distance traveled by the ball, we need to sum up the distances traveled during each downward and upward movement.
16 + 12.96 + 12.96 + 10.5136 + 10.5136 + ... (continue the pattern until the height becomes negligible)
Since we don't have specific information about when the height becomes negligible, we cannot calculate an exact value for the total vertical distance traveled by the ball.
To find the total vertical distance traveled by the ball, we need to consider both the distance it travels when dropped and when it rebounds.
Let's break down the problem step by step:
1. First, calculate the distance the ball travels when dropped from a height of 16 feet.
- The initial distance dropped is 16 feet.
2. Next, calculate the distance the ball rebounds when it hits the ground.
- According to the problem statement, each time the ball rebounds, it travels 0.81 times the height it fell from.
- Therefore, the distance the ball rebounds is 0.81 times the distance it fell.
3. Repeat the process for each subsequent drop and rebound until the distance becomes negligible.
- Each time the ball rebounds, it will travel 0.81 times the distance it fell.
- Continue calculating the distance it rebounds until the distance becomes too small to consider.
4. Add up the distances traveled during each drop and each rebound to find the total vertical distance traveled by the ball.
- Sum up the initial drop distance and all the rebound distances.
Note: Since the problem does not specify any constraints on the number of rebounds or a threshold for when the distance becomes negligible, we assume that the process continues indefinitely until the distance is negligible in practice.
Keep in mind that these calculations involve an infinite geometric series since the distance traveled during rebounds forms a geometric sequence with a common ratio of 0.81. Thus, the total distance can be expressed using a formula for the sum of an infinite geometric series.
Let me calculate the answer for you.