Answer: a. Write a rule for the height of the ball after the Nth bounce.
Step-by-step explanation:
Height(N) = Initial height * (bounce factor)^(N-1)
Where:
- Initial height is the height from which the ball is dropped (in this case, 100 meters).
- Bounce factor is the percentage of the previous height that the ball reaches after each bounce (in this case, 70% or 0.7).
- N is the number of bounces.
b. To find the total vertical distance the ball has traveled when it hits the ground for the 7th time, we need to sum up the heights of each bounce up to the 7th bounce.
Using the formula from part a, we can calculate the height of the ball after each bounce:
Height(1) = 100 meters
Height(2) = 100 * 0.7 = 70 meters
Height(3) = 70 * 0.7 = 49 meters
Height(4) = 49 * 0.7 = 34.3 meters
Height(5) = 34.3 * 0.7 = 24.01 meters
Height(6) = 24.01 * 0.7 = 16.807 meters
Height(7) = 16.807 * 0.7 = 11.7649 meters
To find the total vertical distance, we add up the heights of each bounce:
Total distance = Height(1) + Height(2) + Height(3) + Height(4) + Height(5) + Height(6) + Height(7)
Total distance = 100 + 70 + 49 + 34.3 + 24.01 + 16.807 + 11.7649
Total distance ≈ 305.8819 meters (rounded to 2 decimal places)
c. When the ball stops bouncing, it reaches a height close to zero but not exactly zero due to the rounding of the bounce factor. In this case, we can approximate the total vertical distance of the ball when it stops bouncing as the sum of the heights of all the bounces.
Using the formula from part a, we can calculate the height of the ball after each bounce:
Height(1) = 100 meters
Height(2) = 100 * 0.7 = 70 meters
Height(3) = 70 * 0.7 = 49 meters
Height(4) = 49 * 0.7 = 34.3 meters
...
Height(N) = 100 * (0.7)^(N-1)
To find the total vertical distance when the ball stops bouncing, we can use the formula for the sum of a geometric series:
Total distance = Initial height * (1 - bounce factor^N) / (1 - bounce factor)
Using the given values, we can substitute them into the formula:
Total distance = 100 * (1 - 0.7^N) / (1 - 0.7)
Since we want to find the total vertical distance when the ball stops bouncing, we can take the limit as N approaches infinity:
Total distance = 100 / (1 - 0.7) = 100 / 0.3 = 333.33 meters (rounded to 2 decimal places)
Therefore, the total vertical distance of the ball when it stops bouncing is approximately 333.33 meters.