Question

A Super Happy Fun Ball is dropped from a height of 9 feet and rebounds 6/7 of the distance from which it fell.

How many times will it bounce before its rebound is less than 1 foot?

How far will the ball travel before it comes to rest on the ground?

Answer 1

1) Let it will bounce for n times.

Then, 9*(6/7)ⁿ < 1

i.e., (6/7)ⁿ < 1/9

Taking log on both sides, we get,

i.e., n*log(6/7) < log(1/9)

i.e., -n*log(7/6) < -log(9)

i.e., n*log(7/6) > log(9)

i.e., n > log(9)/log(7/6)

i.e., n > 14.25374562

Since, n is integer. Therefore, n = 15.

Hence, it will bounce 15 times before its rebound is less than 1 foot.

2) The ball will travel = {9+2[9*(6/7)+9*(6/7)²+9*(6/7)³+9*(6/7)⁴+..........]} feet

= {2*9*[1+(6/7)+(6/7)²+(6/7)³+(6/7)⁴+..........] - 9} feet

= {[18*1/{1-(6/7)}] - 9} feet

= [(18*7) - 9] feet

= [126 - 9] feet

= 117 feet

Therefore, the ball will travel 117 feet before it comes to rest on the ground.