Suppose that a ball has a rebound percentage of 75 and the ball is dropped from a height of 5ft onto the concrete below. Make a table of the data.
Write a recursive rule for the data.
Write a closed-form function rule for the data.
What would the height of the 5th bounce of this ball be if the initial drop height is 10 ft above ground?
To create a table of data for the ball's rebounds, we can use the given rebound percentage of 75. Each time the ball bounces, its height will be multiplied by 75% (0.75) of its previous height.
Let's start with an initial drop height of 5ft:
| Bounce | Height (ft) |
|--------|-------------|
| 0 | 5 |
| 1 | 5 * 0.75 |
| 2 | (5 * 0.75) * 0.75 |
| 3 | ((5 * 0.75) * 0.75) * 0.75 |
| 4 | (((5 * 0.75) * 0.75) * 0.75) * 0.75 |
To write a recursive rule for the data, we can express the height of bounce n as a function of the height of the previous bounce (n-1). Let's denote the height of the nth bounce as H(n) and the height of the previous bounce as H(n-1).
H(n) = H(n-1) * 0.75
This means that to find the height of the nth bounce, we can simply multiply the height of the previous bounce by 0.75.
To write a closed-form function rule, we can express the height of the nth bounce as a function of the initial drop height (H0) and the number of bounces (n).
H(n) = H0 * (0.75)^n
In this formula, H(n) represents the height of the nth bounce, H0 represents the initial drop height, and (0.75)^n represents the rebound percentage raised to the power of the number of bounces.
To find the height of the 5th bounce if the initial drop height is 10ft above the ground, we can substitute the values into the closed-form function rule:
H(5) = 10 * (0.75)^5
Calculating this, we have:
H(5) = 10 * 0.2373
H(5) ≈ 2.373 ft
Therefore, the height of the 5th bounce would be approximately 2.373 ft.