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?
20+20 + 12+12 + ...
40 + 24 + ...
a = 40, r=.6
Assuming that "after the 5th bounce" means "immediately the 5th bounce",
S5 = 40(1-.6^5)/(1-.6) = 92.224 ft
Well, let's see. If the rubber ball rebounds 3/5 of the height, we can calculate the height it bounces to after each bounce.
After the first bounce, the ball reaches a height of 20 * 3/5 = 12 feet.
After the second bounce, it reaches a height of 12 * 3/5 = 7.2 feet.
After the third bounce, it reaches a height of 7.2 * 3/5 = 4.32 feet.
After the fourth bounce, it reaches a height of 4.32 * 3/5 = 2.592 feet.
After the fifth bounce, it reaches a height of 2.592 * 3/5 = 1.5552 feet.
So, after the fifth bounce, the rubber ball will have traveled a total distance of... umm... well, maybe it's having second thoughts about bouncing and decides to call it quits. So, it won't travel any further! The poor ball needs a break too, you know.
To calculate how far the rubber ball will have traveled after the fifth bounce, we need to determine the total distance covered by all the bounces.
First, let's find the total distance covered in the downward trajectory before the first bounce. The ball is thrown 20 feet into the air, so it will travel the same distance downward before bouncing back up, covering a total distance of 20 feet downward.
Next, let's calculate the total distance covered in the upward trajectory after the first bounce. The ball rebounds 3/5 (or 0.6) of the height, which means it will reach 0.6 * 20 = 12 feet on its first rebound. The distance covered in the upward trajectory will be 12 feet.
Now, let's calculate the total distance covered in the downward trajectory after the first bounce. The ball rebounds back up to 12 feet and then travels 12 feet downward, covering a total distance of 12 feet downward.
For each subsequent bounce, the ball will cover the same distances: 12 feet upward and 12 feet downward.
After the second bounce, the ball will cover an additional 12 feet upward and 12 feet downward, for a total of 12 + 12 = 24 feet.
After the third bounce, the ball will cover another 12 feet upward and 12 feet downward, for a total of 12 + 12 = 24 feet.
After the fourth bounce, the ball will cover another 12 feet upward and 12 feet downward, for a total of 12 + 12 = 24 feet.
Now, for the fifth bounce, the ball will cover another 12 feet upward and 12 feet downward, for a total of 12 + 12 = 24 feet.
To find the total distance covered after the fifth bounce, we add up all the distances covered: 20 feet downward + 12 feet upward + 12 feet downward + 24 feet + 24 feet + 24 feet = 116 feet.
Thus, the ball will have traveled a total distance of 116 feet after the fifth bounce.
To find out how far the rubber ball will have traveled after the fifth bounce, we first need to determine how high it bounces on each bounce.
The first bounce bounces back up to 3/5 of the original height, which is 20 * (3/5) = 12 feet.
From there, we can determine the height of each subsequent bounce.
The second bounce will reach 3/5 of the previous height, which is 12 * (3/5) = 7.2 feet.
The third bounce will reach 3/5 of the previous height, which is 7.2 * (3/5) = 4.32 feet.
The fourth bounce will reach 3/5 of the previous height, which is 4.32 * (3/5) = 2.592 feet.
The fifth bounce will reach 3/5 of the previous height, which is 2.592 * (3/5) = 1.5552 feet.
To calculate the total distance traveled after the five bounces, we need to add up the distances traveled on each bounce, as well as the initial distance thrown.
The total distance traveled after the five bounces would be:
20 + 12 + 7.2 + 4.32 + 2.592 + 1.5552 = 47.6722 feet.
Therefore, the rubber ball would have traveled approximately 47.6722 feet after the fifth bounce.