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A rubber ball is dropped from a height of 50 inches. On each rebound, the ball rises to 3/4 of the height from which it last fell. Find the distance travelled by the ball in coming to rest.

Solution:

The ball dropped from 50 inches then rises at 0.75 of 50, then falls. It rises again at 0.75 of 0.75 of 50, then repeats the pattern.

50, 0.75 (50), 0.75(0.75)(50), 0.75(0.75)(0.75)(50),... This is a pattern for a geometric series

Since this is infinite, the sum of all fall is $$s = \frac{a}{1-r}$$, where $$r = 0.75$$ and $$a = 50$$

$$s = \frac{50}{1-0.75} = 200$$

Sum of all rises, is the same as the sum of all fall except 50, thus sum of all rises is 150

The total distance travelled by the rubber ball is 200+150 = 350 inches