Question

A ball recovers 80% of its initial height each time after it bounces off of the floor. If the ball is
dropped from a height of 12 feet above the floor, find the total distance traveled in the air by the ball
before it comes to rest.

Answer 1

looks kind of like this one
http://openstudy.com/study?login#/updates/4fb99611e4b05565342e88f7
but there is a slight difference here

Answer 2

we are going to sum a geometric series, but we have to be careful
first the ball drops 12 feet
then comes up 80% of 12 which is \(12\times .8\) and then it goes down the same distance, so we have on the first bounce \(12\times .8+12\times .8=24\times .8\)

Answer 3

on the next bounce it comes up \(12\times .8^2\) and also goes down \(12\times .8^2\) for a total of \(24\times .8^2\) similarly on the next bounce up \(12\times .8^3\) and down \(12\times .8^3\) for a total of \(24\times .8^3\) so we will add up the following
\[12+24\times .8+24\times .8^2+24\times .8^3+...\] and we have to ignore the first 12 when we add the geometric series, because that is not part of it

Answer 4

to add
\[24(.8+.8^2+.8^3+...\] use
\[24\times \frac{.8}{1-.8}\] then add the 12 at the end

Answer 5

if you get stuck let me know