The ball reach after bouncing four time? how many bounces does it take for the ball to reach a height of less than one inch?

After the 'n'th bounce, the ball returns to a maximum height of

(100) times (0.75) ^n.

- - After 4 bounces, the maximum height is (100) (0.75) ^4 = 31.64 inches.

- - To look for the maximum height of 1 inch, (100) (0.75) ^x = 1

Let's take the log of each side: log (100) + x log (0.75) = 0

Subtract log (100) from each side: x log (0.75) = - log (100)

Divide each side by log (0.75) : x = - log (100) / log (0.75).

log (100) = 2. So ... x = - 2 / log (0.75) = - 2 / - 0.124939 = 16.00785

So the ball returns to a slim hair more than 1 inch high on the 16th bounce,

and returns to definitely less than 1 inch high on the 17th bounce.