Investigating Patterns and Recursion

Question

Kierra drops a rubber ball from the rooftop observatory of a building to a concrete landing at ground level. The height of the building is 264 square feet. As the ball bounces, Kierra estimates that the ball rebounds to 2/3 the height of the previous bounce. Assuming the ball continues to bounce indefinitely, use a convergent infinite series to approximate the total difference the ball travels.

Answer 1

Hi Gaurav,

What a tricky problem! There's a lot of steps involved, and I hope I did a good enough job explaining each of them. Good luck with similar problems!

Ross

Answer 2

I'm going to make a couple of assumptions. First, that the height of the building is 264 feet (instead of square feet). And second, that what is being asked for is he total distance the ball travels.

The initial distance is 264, then on the bounce it is 2/3 of that, then on the next bounce it is 2/3 again, etc. So the sequence is 264, 176, 117.3, 78.2, 52.1, . . .

The series is the sum of all those (and all continued) terms so 264 + 176 + 117.3 + 78.2 + 52.1 + . . .

An infinite series like this can be calculated using the equation:

S_∞ = a₁/(1 - r) where a1 is the first term and r is the multiplier (in this case 2/3). So:

S_∞ = 264/(1 - 2/3) = 264/(1/3) = 264*3 = 792 feet