A ball is dropped from the height of 20m. It bounces 60% of its previous height each bounce. The total distance travelled is made of the upward and downward bounces. What is the total vertical distance the ball has travelled after the sixth bounce?

A geometric series with a=40 (for a round trip) and r = 0.6

S6 = 40(1-.6^6)/(1-.6) = 95.33

Now subtract 20 from that, since the 1st "bounce" started from the top.

To find the total vertical distance the ball has traveled after the sixth bounce, we need to calculate the distance traveled for each bounce and sum them up.

We know that the ball bounces up to 60% of its previous height each time. Let's calculate the distance traveled for each bounce:

First bounce:
The ball is dropped from a height of 20m, so it first travels 20m downwards. Then, it bounces up to 60% of its previous height.
Distance traveled = 20m (downward) + 0.6 * 20m (upward)
Distance traveled = 20m + 12m
Distance traveled = 32m (up and down)

Second bounce:
The ball is now bouncing from a height of 12m (60% of 20m). It travels this distance downwards, then bounces up to 60% of its previous height.
Distance traveled = 12m (downward) + 0.6 * 12m (upward)
Distance traveled = 12m + 7.2m
Distance traveled = 19.2m (up and down)

We can continue this calculation for the next four bounces:

Third bounce:
Height = 0.6 * 12m = 7.2m
Distance = 7.2m (downward) + 0.6 * 7.2m (upward) = 7.2m + 4.32m = 11.52m (up and down)

Fourth bounce:
Height = 0.6 * 7.2m = 4.32m
Distance = 4.32m (downward) + 0.6 * 4.32m (upward) = 4.32m + 2.592m = 6.912m (up and down)

Fifth bounce:
Height = 0.6 * 4.32m = 2.592m
Distance = 2.592m (downward) + 0.6 * 2.592m (upward) = 2.592m + 1.5552m = 4.1472m (up and down)

Sixth bounce:
Height = 0.6 * 2.592m = 1.5552m
Distance = 1.5552m (downward) + 0.6 * 1.5552m (upward) = 1.5552m + 0.93312m = 2.48832m (up and down)

Finally, we can sum up all the distances traveled for each bounce from 1st to 6th:
Total distance = 32m + 19.2m + 11.52m + 6.912m + 4.1472m + 2.48832m
Total distance ≈ 76.27152m

Therefore, after the sixth bounce, the total vertical distance the ball has traveled is approximately 76.27152 meters.