If the ball dropped from 12 feet and bounces up 2/3 as high on each successive bounce, what is the height of the fourth bounce?

2 10/27 ft . . . . or . . . . 2 feet 4 4/9 inches

Step-by-step explanation:

The height on the first bounce is ...

... (12 ft) × 2/3 = 8 ft

The height on the second bounce is ...

... (8 ft) × 2/3 = 16/3 ft

The height on the third bounce is ...

... (16/3 ft) × 2/3 = 32/9 ft

The height on the fourth bounce is ...

... (32/9 ft) × 2/3 = 64/27 ft = 2 10/27 ft

Conversion of fractional feet to inches

10/27 ft = (12 in/ft) × (10/27 ft) = 40/9 in = 4 4/9 in

Explicit formula

We can see that for each bounce (including the first), we multiply the initial height by 2/3. After n bounces, we have multiplied the initial height by (2/3)^n. Thus the height after n bounces will be ...

... h(n) = (12 ft)×(2/3)^n

For the 4th bounce, this is ...

... h(4) = (12 ft)×(2/3)^4 = (12 ft)×(16/81) = 64/27 ft

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