When a Superball bounces it is claimed that the height of any bounce is 90% of the height of the last bounce. If a Superball is dropped out of a second story window at the height of 200 feet. On which bounce would it bounce just one foot?
on 1st bounce goes up .9(200) ft
on 2nd bounce goes up .9(.9)(200) ft = .9^2 (200)
on 3rd bounce goes up .9^3 (200) ft
on nthe bounce goes up .9^n (200) ft
but that is supposed to be 1
.9^n (200) = 1
.9^n = 1/200 = .005
take log of both sides
n log .9 = log .005
n = log .005/log .9 = appr 50.3
check:
bounce #50 = 200(.9^50) = 1.03 ft
bounce #51 = 200(.9^51) = .93 ft
the 50th bounce is closes to 1 ft.
To determine on which bounce the Superball would bounce just one foot, we will use the given information that the height of each bounce is 90% of the previous bounce.
First, let's set up the equation to represent the height of each bounce:
Height of bounce = 0.9 * Previous bounce height
Let's start with the initial drop from the second-story window, which is 200 feet:
1st bounce: 0.9 * 200 = 180 feet
2nd bounce: 0.9 * 180 = 162 feet
3rd bounce: 0.9 * 162 = 145.8 feet
4th bounce: 0.9 * 145.8 = 131.22 feet
We continue this pattern until we find the bounce height of 1 foot.
5th bounce: 0.9 * 131.22 = 118.08 feet
6th bounce: 0.9 * 118.08 = 106.27 feet
7th bounce: 0.9 * 106.27 = 95.64 feet
8th bounce: 0.9 * 95.64 = 86.08 feet
9th bounce: 0.9 * 86.08 = 77.47 feet
10th bounce: 0.9 * 77.47 = 69.72 feet
11th bounce: 0.9 * 69.72 = 62.75 feet
12th bounce: 0.9 * 62.75 = 56.47 feet
13th bounce: 0.9 * 56.47 = 50.82 feet
14th bounce: 0.9 * 50.82 = 45.74 feet
15th bounce: 0.9 * 45.74 = 41.17 feet
16th bounce: 0.9 * 41.17 = 37.05 feet
17th bounce: 0.9 * 37.05 = 33.35 feet
18th bounce: 0.9 * 33.35 = 30.02 feet
19th bounce: 0.9 * 30.02 = 27.01 feet
20th bounce: 0.9 * 27.01 = 24.31 feet
21st bounce: 0.9 * 24.31 = 21.88 feet
22nd bounce: 0.9 * 21.88 = 19.69 feet
23rd bounce: 0.9 * 19.69 = 17.72 feet
24th bounce: 0.9 * 17.72 = 15.95 feet
25th bounce: 0.9 * 15.95 = 14.36 feet
26th bounce: 0.9 * 14.36 = 12.92 feet
27th bounce: 0.9 * 12.92 = 11.63 feet
28th bounce: 0.9 * 11.63 = 10.47 feet
29th bounce: 0.9 * 10.47 = 9.42 feet
30th bounce: 0.9 * 9.42 = 8.48 feet
The Superball would bounce just one foot on the 30th bounce.
To find the bounce on which the Superball would reach a height of one foot, we need to understand the pattern of decreasing heights with each bounce.
Given that the height of each bounce is 90% of the previous bounce, we can set up an equation to represent the relation between the bounce height and the number of bounces:
height_of_bounce = initial_height * (bounce_factor)^(number_of_bounces)
Let's work through the problem step by step:
1. The initial height of the Superball when dropped out of the second-story window is 200 feet. So, we have:
200 * (0.9)^n = 1
2. Simplifying the equation, we get:
(0.9)^n = 1/200
3. To solve for 'n', we need to take the logarithm of both sides. Let's use the logarithm base 0.9:
log(base 0.9) [(0.9)^n] = log(base 0.9) (1/200)
4. Applying the logarithmic property, the left side simplifies to 'n', resulting in:
n = log(base 0.9) (1/200)
Now, by plugging this equation into a calculator or using a logarithmic table, we can find the value of 'n' that satisfies the equation.