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Find the distance traveled by a bouncing ball dropped from a height of 80 cm that returns to 1/4 of its previous height after each bounce. Round your answer to one decimal place.

User Arild
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1 Answer

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The total vertical distance that the ball travels is 84m.

This scenario involves an infinite geometric series! We can separate the ball's movement into two series: one for its downward falls and another for its upward bounces.

Downward Falls:

Starting from 12 m, each fall is 3/4 of the previous height. This forms a geometric series with:

First term (a) = 12 m

Common ratio (r) = 3/4

Upward Bounces:

Starting from 9 m (3/4 of 12 m), each bounce is also 3/4 of the previous height. This forms another geometric series with:

First term (a) = 9 m

Common ratio (r) = 3/4

Now, we use the formula for the sum of an infinite geometric series: S = a / (1 - r).

Downward Distance:

S_down = 12 / (1 - 3/4) = 48 m

Upward Distance:

S_up = 9 / (1 - 3/4) = 36 m

Total Distance:

To find the total vertical distance, we add the distance traveled in both directions:

Total Distance = S_down + S_up = 48 m + 36 m = 84 m

Therefore, the ball travels a total vertical distance of 84 meters before coming to rest.

Question:

A ball is dropped from a height of 12m. After each bounce, it rises to 3/4 of the height of the previous bounce. What is the total vertical distance that the ball travels?

User Karsten
by
8.0k points
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