Problem 55
Question
A ball is dropped from a height of 8 feet. On each bounce, it rises to half its previous height. When the ball hits the ground for the seventh time, how far has it traveled?
Step-by-Step Solution
Verified Answer
Answer: The total distance the ball has traveled when it hits the ground for the seventh time is 23.75 feet.
1Step 1: Identify the geometric series terms
The ball is first dropped from a height of 8 feet, which is our first term (a). Each consecutive term is half the previous term, which means the common ratio (r) is 0.5.
2Step 2: Calculate the downward distance
We want to find the sum of distances the ball falls down in 7 bounces. This forms a geometric series with a = 8 and r = 0.5.
Using the formula for the sum of the first n terms of a geometric series:
\(S_n = \frac{a(1 - r^n)}{1 - r}\)
For n = 7 (the ball hits the ground 7 times):
\(S_7 = \frac{8(1 - (0.5)^7)}{1 - 0.5}\)
\(S_7 = \frac{8 (1 - 0.0078125)}{0.5}\)
\(S_7 = \frac{8 (0.9921875)}{0.5}\)
Downward distance = \(S_7 = 15.875\) feet
3Step 3: Calculate the upward distance
The upward distance is essentially the same as the downward distance, but we must exclude the first term as the ball does not bounce upward initially. Therefore, n = 6 for the upward distance:
\(S_6 = \frac{8(1 - (0.5)^6)}{1 - 0.5} - 8\)
\(S_6 = \frac{8 (1 - 0.015625)}{0.5} - 8\)
\(S_6 = \frac{8 (0.984375)}{0.5} - 8\)
Upward distance = \(S_6 = 7.875\) feet
4Step 4: Calculate the total distance traveled
Finally, sum the downward and upward distances to get the total distance traveled:
Total distance = Downward distance + Upward distance
Total distance = 15.875 feet + 7.875 feet
Total distance = 23.75 feet
When the ball hits the ground for the seventh time, it has traveled 23.75 feet.
Key Concepts
Geometric SequenceSum of Geometric SeriesExponential DecaySeries Convergence
Geometric Sequence
Understanding a geometric sequence is fundamental in solving problems like the bouncing ball exercise. A geometric sequence is a series of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
For example, if we have a sequence starting with 1 and have a common ratio of 2, our sequence will be 1, 2, 4, 8, and so on. The ball's height in the exercise forms a geometric sequence starting from 8 feet, with each bounce rising to half of its previous height, leading to a common ratio of 0.5.
For example, if we have a sequence starting with 1 and have a common ratio of 2, our sequence will be 1, 2, 4, 8, and so on. The ball's height in the exercise forms a geometric sequence starting from 8 feet, with each bounce rising to half of its previous height, leading to a common ratio of 0.5.
Sum of Geometric Series
To find the total distance traveled by the ball, we needed to calculate the sum of geometric series. In a geometric series, each term is the product of the previous term and the common ratio. The sum of the first 'n' terms can be calculated using the formula:
\[S_n = \frac{a(1 - r^n)}{1 - r}\]
where Sn is the sum of the series up to the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Using this formula, we determined the ball's downward and upward distances separately. This calculation is important for exercises like ours where the sum of a series up to a certain term is required.
\[S_n = \frac{a(1 - r^n)}{1 - r}\]
where Sn is the sum of the series up to the nth term, 'a' is the first term, 'r' is the common ratio, and 'n' is the number of terms.
Using this formula, we determined the ball's downward and upward distances separately. This calculation is important for exercises like ours where the sum of a series up to a certain term is required.
Exponential Decay
Exponential decay refers to a decrease that happens proportionally to the current value. In other words, as the value decreases, the rate of decline slows down. This concept is exemplified by the bounce heights of the ball.
In our exercise, each bounce height is half of the previous bounce, demonstrating an exponential decay. This decay pattern can also be observed in other real-life scenarios such as radioactive decay or depreciation of assets over time.
In our exercise, each bounce height is half of the previous bounce, demonstrating an exponential decay. This decay pattern can also be observed in other real-life scenarios such as radioactive decay or depreciation of assets over time.
Series Convergence
The concept of series convergence is applicable when we are considering an infinite number of terms in a series. A series converges if the terms approach a specific value as more terms are added.
In the case of the bouncing ball, as the number of bounces approaches infinity, the total vertical distance would converge to a finite value. However, in this exercise, we only need to consider the distance covered in seven bounces, but if asked about the infinite bounces, the sum would still be finite due to the nature of a converging geometric series.
In the case of the bouncing ball, as the number of bounces approaches infinity, the total vertical distance would converge to a finite value. However, in this exercise, we only need to consider the distance covered in seven bounces, but if asked about the infinite bounces, the sum would still be finite due to the nature of a converging geometric series.
Other exercises in this chapter
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Use the Binomial Theorem to factor the expression. $$16 z^{4}+32 z^{3}+24 z^{2}+8 z+1$$
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