$\"\"$ \ \

\

The ball is dropped from a height of $\"\"$ meters. \ \

\

On each bounce, the ball rises to $\"\"$ of the height it reached on the previous bounce.

\

(a) Find the total vertical distance the ball travells.

\

Hence, $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

Therefore, an infinite sequence that can be used to represent this situation is

\

$\"\"$

\

The series as the sum of the two infinite geometric series:

\

one series represents the distance the ball travels when falling.

\

Another series that represents the distance the ball travels when bouncing back up.

\

Therefore, the two series are $\"\"$ and $\"\"$ .

\

The sum of infinite geometric series one is $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ .

\

$\"\"$ .

\

The sum of infinite geometric series two is $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ .

\

$\"\"$ .

\

Therefore, the total vertical distance the ball travells is $\"\"$ .

\

$\"\"$

\

(b)

\

The ball makes its first complete bounce is $\"\"$ . \ \

\

Each complete bounce that follows takes $\"\"$ times as long as the preceding bounce.

\

Hence, $\"\"$

\

Estimate the total amount of time.

\

The sum of infinite geometric series is $\"\"$ .

\

Substitute $\"\"$ and $\"\"$ .

\

$\"\"$ .

\

Therefore, the total amount of time that the ball bounces is $\"\"$ .

\

$\"\"$

\

(a) The total vertical distance the ball travells is $\"\"$ .

\

(b) The total amount of time that the ball bounces is $\"\"$ .