A rubber ball is thrown 20 feet into the air. If it rebounds 3/5 of the height, how far will it have traveled after the fifth bounce?
To determine the distance the ball will have traveled after the fifth bounce, we need to calculate the total distance traveled by the ball on each bounce.
First, let's calculate the height of the ball after each bounce. We know that the ball rebounds 3/5 of its previous height.
On the first bounce:
Height = 20 feet
On the second bounce:
Height = (3/5) * 20 feet
On the third bounce:
Height = (3/5) * (3/5) * 20 feet
On the fourth bounce:
Height = (3/5) * (3/5) * (3/5) * 20 feet
On the fifth bounce:
Height = (3/5) * (3/5) * (3/5) * (3/5) * 20 feet
Now, let's calculate the total distance traveled by the ball on each bounce, assuming the ball covers twice the height on each bounce (up and down):
On the first bounce:
Distance = 2 * 20 feet
On the second bounce:
Distance = 2 * (2 * (3/5) * 20 feet)
On the third bounce:
Distance = 2 * (2 * (3/5) * (3/5) * 20 feet)
On the fourth bounce:
Distance = 2 * (2 * (3/5) * (3/5) * (3/5) * 20 feet)
On the fifth bounce:
Distance = 2 * (2 * (3/5) * (3/5) * (3/5) * (3/5) * 20 feet)
Now, we can simplify the calculations:
On the first bounce:
Distance = 40 feet
On the second bounce:
Distance = 2 * (2 * (3/5) * 20 feet) = 2 * (2 * 12 feet) = 2 * 24 feet = 48 feet
On the third bounce:
Distance = 2 * (2 * (3/5) * (3/5) * 20 feet) = 2 * (2 * 12 * 9/25 feet) = 2 * (216/25 feet) = 2 * 8.64 feet = 17.28 feet
On the fourth bounce:
Distance = 2 * (2 * (3/5) * (3/5) * (3/5) * 20 feet) = 2 * (2 * 12 * 9/25 * 9/25 feet) = 2 * (1944/625 feet) = 2 * 3.11 feet = 6.22 feet
On the fifth bounce:
Distance = 2 * (2 * (3/5) * (3/5) * (3/5) * (3/5) * 20 feet) = 2 * (2 * 12 * 9/25 * 9/25 * 9/25 feet) = 2 * (17496/15625 feet) = 2 * 2.8 feet = 5.6 feet
Therefore, the ball will have traveled a total distance of 40 + 48 + 17.28 + 6.22 + 5.6 feet after the fifth bounce, which is approximately 117.1 feet.
To find out how far the rubber ball will have traveled after the fifth bounce, we need to first determine the total distance covered during a single bounce.
Given that the rubber ball is thrown 20 feet into the air and rebounds 3/5 of the height, we can calculate the distance covered in a single bounce.
First, let's calculate the height reached after the first bounce. The ball rebounds 3/5 of the original height, so after the first bounce, it reaches a height of (3/5) * 20 feet = 12 feet.
Now, we have the distance covered during the ascent (20 feet) and the descent (12 feet) in the first bounce. So, the total distance covered during a single bounce is 20 feet + 12 feet = 32 feet.
To find out the total distance covered after the fifth bounce, we need to calculate the cumulative distance.
The total distance covered after one bounce is 32 feet. After the second bounce, the distance covered doubles from the first bounce, so it is 32 feet * 2 = 64 feet.
After the third bounce, the distance covered doubles again from the second bounce, so it is 64 feet * 2 = 128 feet.
After the fourth bounce, the distance covered doubles again from the third bounce, so it is 128 feet * 2 = 256 feet.
Finally, after the fifth bounce, the distance covered doubles from the fourth bounce, so it is 256 feet * 2 = 512 feet.
Therefore, the rubber ball will have traveled 512 feet after the fifth bounce.