Answer:
Total distance traveled vertically by the ball at the 22nd time it hits ground is the sum of the upward travel and the downward travel which equals to 230.54 feet. Both are geometric progressions.
Explanation:
(1) Downward travel:
Initial height = a = 81 feet
On each impact to the ground, ball rebounds 48 % of its previous height, so:
Sum of 22 times it travels downward = a * [ { 1 - (0.48^22) } / ( 1 - 0.48 ) ]
= 81 * 1.92307
= 155.77 feet
(2) Upward (rebound) travel:
Initial height = a = 81 feet
Height after 1st rebound = b = 81 * 0.48 = 38.88 feet
Since the ball will rebound only 21 times, so we will calculate only distance of only 21 rebounds:
Sum of 21 times it travels upward = b * [ { 1 - (0.48^21) } / ( 1 - 0.48 ) ]
= 38.88 * 1.923
= 74.77 feet
So the total travel = 155.77 + 74.77 = 230.54 feet approximately