Problema Solution
a certain ball is dropped a height of x feet, it always bounces up to 2/3x feet. Suppose the ball is dropped from 10 feet and is caught exactly when it touches the ground after the 30th bounce, what is the total distance traveled by the ball? express answer in exponential notation.
Answer provided by our tutors
before the 1st bounce the ball travels 10 ft
between the 1st and 2nd bounce it travels: 2*10(2/3)
between the 2nd and 3rd bounce it travels: 2*10(2/3)^2
....
between the 29th and 30th bounce travels: 2*10(2/3)^29
lets add all of the distances:
10 + 2*10(2/3) + 2*10(2/3)^2 + ...+2*10(2/3)^29 =
= 10 + 20((2/3) + (2/3)^2 + ... + (2/3)^29)) =
= 10 + 20(2*(1 - (2/3)^30)) = 10 + 40 - 40(2/3)^30 = 50 ft approximately
the sum of the geometric series (2/3) + (2/3)^2 + ... + (2/3)^29 is:
(2/3) + (2/3)^2 + ... + (2/3)^29 = (2/3)(1 - (2/3)^30)/(1 - 2/3) = 2*(1 - (2/3)^30)
the total distance traveled by the ball is 50 ft approximately.