Math if a rubber ball drops from a height of 28 meters, and on each bounce it rebounds up 30% of its previous height, how far will it traveled at the moment it hits the ground for the 16th time
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To find out how far the rubber ball will travel at the moment it hits the ground for the 16th time, we need to calculate the total distance traveled during all the bounces.
First, we know that the ball is dropped from a height of 28 meters. We can consider this as the initial drop. To find the total distance traveled, we need to sum up the distances covered during each bounce.
Let's break down the problem into individual bounces and calculate the distance traveled for each one.
1st bounce: The ball reaches a maximum height of 28 meters, and then it rebounds 30% of that height, which is (30/100) * 28 = 8.4 meters. The total distance traveled after the 1st bounce is 28 + 8.4 = 36.4 meters.
2nd bounce: The ball rebounds from 8.4 meters and reaches a maximum height of (30/100) * 8.4 = 2.52 meters. The total distance traveled after the 2nd bounce is 36.4 + 2.52 = 38.92 meters.
3rd bounce: The ball rebounds from 2.52 meters and reaches a maximum height of (30/100) * 2.52 = 0.756 meters. The total distance traveled after the 3rd bounce is 38.92 + 0.756 = 39.676 meters.
To find the distance traveled after the 16th bounce, we continue this pattern, calculating the maximum height reached and adding it to the previous total distance traveled.
16th bounce: We need to calculate the maximum height reached on the 15th bounce and then find the maximum height on the 16th bounce and add it to the total distance traveled after the 15th bounce.
To make this calculation easier, we can use a formula:
Distance = Initial height + (1 + rebound percentage)^bounce number
Using this formula:
Distance after the 15th bounce = 28 + (1 + 0.3)^15 = 28 + (1.3)^15 = 28 + 38.755074 = 66.755074 meters (rounded to 6 decimal places).
Now we can calculate the distance traveled after the 16th bounce:
Distance after the 16th bounce = 66.755074 + (1 + 0.3)^16 = 66.755074 + (1.3)^16 = 66.755074 + 50.287189 = 117.042263 meters (rounded to 6 decimal places).
Therefore, the rubber ball will travel approximately 117.042263 meters at the moment it hits the ground for the 16th time.