Question

A ball is dropped form a height of 100 inches. The ball bounces, each time reaching a lower and lower height. The height of each bounce is 75% of the previous bounce. What height will the ball reach after bouncing four times? How many bounces does it take for the ball to reach a height of less than one inch?

Answer 1

75% of a number is as much as 3/4 of the same number, so:

1st bounce: 3/4 * 100 = 75
2nd bounce: 3/4 * 75 = 56,25
3rd bounce: 3/4 * 56,25 = 42,1875
4th bounce: 3/4 * 42,1875 = 31,640625
5th bounce: 3/4 * 31,640625 = 23,73046875
6th bounce: 3/4 * 23,73046875 = 17,7978515625
7th bounce: 3/4 * 17,7978515625 = 13,348388671875
8th bounce: 3/4 * 13,348388671875 = 10,01129150390625
9th bounce: 3/4 * 10,01129150390625 = 7,508468627929688
10th bounce: 3/4 * 7,508468627929688 = 5,631351470947266
11th bounce: 3/4 * 5,631351470947266 = 4,223513603210449
12th bounce: 3/4 * 4,223513603210449 = 3,167635202407837
13th bounce: 3/4 * 3,167635202407837 = 2,375726401805878
14th bounce: 3/4 * 2,375726401805878 = 1,781794801354408
15th bounce: 3/4 * 1,781794801354408 = 1,336346101015806
16th bounce: 3/4 * 1,336346101015806 = 1,002259575761855
17th bounce: 3/4 * 1,002259575761855 = 0,7516946818213909

Answers:

After bouncing four times, the ball will reach a height of 31,640625 inches.
The ball will reach a height of less than one inch after 17 bounces.

PS: I know it's kind of a walk-around, by it still gives us the answer :)

Answer 2

After the 'n'th bounce, the ball returns to a maximum height of

(100) times (0.75)^n .

-- After 4 bounces, the maximum height is (100) (0.75)^4 = 31.64 inches.

-- To look for the maximum height of 1 inch, (100) (0.75)^x = 1

Let's take the log of each side: log(100) + x log(0.75) = 0

Subtract log(100) from each side: x log(0.75) = -log(100)

Divide each side by log(0.75) : x = -log(100) / log(0.75).

log(100) = 2 . So ... x = -2 / log(0.75) = -2 / -0.124939 = 16.00785

So the ball returns to a slim hair more than 1 inch high on the 16th bounce,
and returns to definitely less than 1 inch high on the 17th bounce.