A ball is dropped from the top of a 25-m ladder. In each bounce, the ball reaches a vertical height
that is 3/5 the previous vertical height.Determine the total vertical distance traveled by the ball
when it contacts the ground for the sixth time. Express your answer to the nearest tenth of a metre.
To determine the total vertical distance traveled by the ball when it contacts the ground for the sixth time, we can use the geometric series formula:
S = a(1 - r^n) / (1 - r)
where:
S is the sum of the geometric series,
a is the first term of the series (which is the initial height of the ball),
r is the common ratio (which is 3/5),
n is the number of terms in the series.
In this case, the initial height of the ball is 25 meters, the common ratio is 3/5, and we want to find the sum of the series when it reaches the ground for the sixth time (n = 6).
Plugging in these values into the formula, we have:
S = 25(1 - (3/5)^6) / (1 - 3/5)
Calculating the value of (3/5)^6, we get:
S = 25(1 - 0.0467) / (1 - 3/5)
Simplifying further:
S = 25(0.9533) / (2/5)
S = 25 * 0.9533 * (5/2)
S = 59.58125
Rounded to the nearest tenth of a meter, the total vertical distance traveled by the ball when it contacts the ground for the sixth time is approximately 59.6 meters.
To determine the total vertical distance traveled by the ball in the given scenario, we need to find the sum of the distances traveled in each bounce until the ball contacts the ground for the sixth time.
Let's break down the problem and solve it step by step:
1. Determine the initial vertical height from which the ball is dropped: The initial vertical height is given as 25 meters since the ball is dropped from the top of a 25-meter ladder.
2. Calculate the vertical distance traveled in each bounce: The ball reaches a vertical height that is 3/5 (or 0.6) of the previous vertical height in each bounce.
3. Count the number of bounces until the ball contacts the ground for the sixth time: Each time the ball hits the ground, it bounces back up. Therefore, the ball contacts the ground one less time than the number of bounces. So, in this case, the ball contacts the ground for the sixth time on its seventh bounce.
4. Calculate the total vertical distance traveled: We need to sum up the distances traveled in each of the six bounces preceding the seventh bounce.
Now, let's perform the calculations:
Initial height = 25 meters
First bounce: 0.6 * 25 = 15 meters
Second bounce: 0.6 * 15 = 9 meters
Third bounce: 0.6 * 9 = 5.4 meters
Fourth bounce: 0.6 * 5.4 = 3.24 meters
Fifth bounce: 0.6 * 3.24 = 1.944 meters
Sixth bounce: 0.6 * 1.944 = 1.1664 meters
Total vertical distance = 25 + 15 + 9 + 5.4 + 3.24 + 1.944 + 1.1664 = 61.74 meters
Therefore, the total vertical distance traveled by the ball when it contacts the ground for the sixth time is approximately 61.7 meters (rounded to the nearest tenth of a meter).