A rubber ball is dropped from a height of 729 feet, and at each bounce it rebounds one-third of the height from which it last fell. How far has the ball travelled by the time it strikes the ground for the 5th time?

first time: 729/3=243

second time: 243/3
third time:81/3
fourth time: 27/3
fifth time: 9/3=3ft

distance for 1st bounce = 729

distance for 2nd bounce = 2(1/3)(729) = 2(243)
distance for 3rd bounce = 2(1/3)(243) = 2(81)
distance for 4th bounce = 2(1/3)(81) = 2(27)
distance for 5th bounce = 2(1/3)(27) = 2(9)

total distance
= 729 + 2(243+81+27+9) = 1449

To find out how far the ball has traveled by the time it strikes the ground for the 5th time, we need to determine the total distance covered during each bounce.

Let's calculate the distance covered during each bounce:

1st bounce: The ball is dropped from a height of 729 feet, so it bounces back up to one-third of 729 feet, which is (1/3) * 729 = 243 feet.

2nd bounce: The ball is dropped from a height of 243 feet, so it bounces back up to one-third of 243 feet, which is (1/3) * 243 = 81 feet.

3rd bounce: The ball is dropped from a height of 81 feet, so it bounces back up to one-third of 81 feet, which is (1/3) * 81 = 27 feet.

4th bounce: The ball is dropped from a height of 27 feet, so it bounces back up to one-third of 27 feet, which is (1/3) * 27 = 9 feet.

5th bounce: The ball is dropped from a height of 9 feet, so it bounces back up to one-third of 9 feet, which is (1/3) * 9 = 3 feet.

Now, let's calculate the total distance covered by summing up the distances covered during each bounce:

Total distance = 729 + 243 + 81 + 27 + 9 + 3

Total distance = 1092 feet

Therefore, the ball has traveled 1092 feet by the time it strikes the ground for the 5th time.

To find out how far the ball has traveled by the time it strikes the ground for the 5th time, we can approach it step by step.

Step 1: Determine the distance the ball travels after each bounce.
The ball is dropped from a height of 729 feet, so after the first bounce, it reaches a height of 729 * (2/3) = 486 feet. The distance traveled in the first bounce is 729 + 486 = 1215 feet.

Step 2: Calculate the total distance traveled by the ball after each bounce.
After the second bounce, the ball reaches a height of 486 * (2/3) = 324 feet. The distance traveled in the second bounce is 486 + 324 = 810 feet.
After the third bounce, the ball reaches a height of 324 * (2/3) = 216 feet. The distance traveled in the third bounce is 324 + 216 = 540 feet.
After the fourth bounce, the ball reaches a height of 216 * (2/3) = 144 feet. The distance traveled in the fourth bounce is 216 + 144 = 360 feet.
After the fifth bounce, the ball reaches a height of 144 * (2/3) = 96 feet. The distance traveled in the fifth bounce is 144 + 96 = 240 feet.

Step 3: Sum up the distances traveled during each bounce.
The total distance traveled by the ball after the first five bounces is 1215 + 810 + 540 + 360 + 240 = 3165 feet.

Therefore, the ball has traveled 3165 feet by the time it strikes the ground for the 5th time.