a bouncy ball rebounds to 90% of the height of the preceding bounce. jason drops a bouncy ball from initial height of 25 feet
(a) write out the sequence of the height of the first 4 bounces.
(b) derive an explicit formula for the rebound height of a bouncy ball dropped from an intial height of 25 feet.
initial height = 25 ft
height after 1 bounce = 25(.9)
height after 2 bounces = 25(9)^2
height after 3 bounces = 25(.9)^3
height after 4 bounces = 25(.9)^4
..
height of ball after nth bounce = 25(.9)^n
(a) To find the sequence of the height of the first 4 bounces, we can use the information that the ball rebounds to 90% of the height of the preceding bounce.
The initial drop: 25 feet
First bounce: 90% of 25 feet = 22.5 feet
Second bounce: 90% of 22.5 feet = 20.25 feet
Third bounce: 90% of 20.25 feet = 18.225 feet
Fourth bounce: 90% of 18.225 feet = 16.4025 feet
So, the sequence of the height of the first 4 bounces is: 25 feet, 22.5 feet, 20.25 feet, 18.225 feet, 16.4025 feet.
(b) To derive an explicit formula for the rebound height of a bouncy ball dropped from an initial height of 25 feet, we can use the fact that the ball rebounds to 90% of the height of the preceding bounce.
Let's call the height of the initial drop h. Then, the height of the first bounce is 0.9h, the height of the second bounce is 0.9(0.9h) = (0.9)^2h, the height of the third bounce is (0.9)^3h, and so on.
So, the explicit formula for the rebound height of a bouncy ball dropped from an initial height of 25 feet is:
H(n) = (0.9)^n * h
where H(n) represents the height of the nth bounce, n represents the bounce number (starting from 0), and h represents the initial height of the drop (25 feet in this case).
To find the sequence of the height of the first 4 bounces, we can use a recursive formula:
(a) The initial height is 25 feet. Let's write out the first few bounces:
1st bounce: 25 feet
2nd bounce: 25 feet * 90% = 25 * 0.9 = 22.5 feet
3rd bounce: 22.5 feet * 90% = 22.5 * 0.9 = 20.25 feet
4th bounce: 20.25 feet * 90% = 20.25 * 0.9 = 18.225 feet
Therefore, the sequence of the first 4 bounces is: 25 feet, 22.5 feet, 20.25 feet, 18.225 feet.
(b) To derive an explicit formula for the rebound height of a bouncy ball dropped from an initial height of 25 feet, we can use the recursive formula and simplify it:
Let's assume the height after the nth bounce is H_n. We know that H_1 = 25 feet.
From the problem, we are told that the ball rebounds to 90% of the height of the preceding bounce, so:
H_n = H_(n-1) * 0.9
Since H_1 = 25 feet, we can write:
H_2 = H_1 * 0.9 = 25 * 0.9 = 22.5 feet
H_3 = H_2 * 0.9 = 22.5 * 0.9 = 20.25 feet
H_4 = H_3 * 0.9 = 20.25 * 0.9 = 18.225 feet
By observing this pattern, we can see that the rebound height of each bounce is decreasing by a factor of 0.9.
Therefore, the explicit formula for the rebound height of a bouncy ball dropped from an initial height of 25 feet is:
H_n = 25 * 0.9^(n-1)
Where n represents the bounce number and H_n represents the height after the nth bounce.