A ball is dropped from a height of 500 meters. The table below shows the height of each bounce, and the heights form a geometric sequence. How high does the ball bounce on the 8th bounce? Round your answer to the nearest meter.
Let's first find the common ratio of the geometric sequence.
The height of the ball after the first bounce is given as 500 meters.
Let's call this height "h".
The height of the ball after the second bounce is given as h*r, where "r" is the common ratio.
Similarly, the height of the ball after the third bounce is h*r*r.
From the information in the table, we can see that each bounce decreases the height by 1/2.
Therefore, the common ratio "r" is 1/2.
To find the height of the ball after the 8th bounce, we need to calculate h*(1/2)^7.
Substituting h = 500, we get 500*(1/2)^7 = 500*(1/128) = 3.90625 meters.
Rounding this to the nearest meter gives us an answer of 4 meters.
Therefore, the ball bounces to a height of 4 meters on the 8th bounce.
To find the height of the ball on the 8th bounce, we need to establish the pattern of the geometric sequence. Let's assume the initial height of the ball is h, and the common ratio (r) is the ratio between successive terms.
From the table, we can see that the heights form a geometric sequence:
First bounce: h
Second bounce: hr
Third bounce: hr^2
...
Eighth bounce: hr^7
We are given the initial height of the ball (h = 500 meters). Now we need to find the common ratio (r).
From the given information, the height of the ball decreases with each bounce. This suggests that the common ratio is less than 1. To find the common ratio, we can divide the height of any two consecutive bounces.
Let's take the second and first bounces: hr / h = r
From this, we can see that r = 0.8.
Now we can find the height of the ball on the 8th bounce:
Eighth bounce: hr^7 = 500 * 0.8^7
Using a calculator, we find that:
hr^7 ≈ 161.051
Therefore, the ball bounces to approximately 161 meters on the 8th bounce.
To find the height of the ball on the 8th bounce, we need to understand how the height changes with each bounce. The problem states that the heights form a geometric sequence, meaning that each term is obtained by multiplying the previous term by a constant factor.
Let's denote the height of the ball on the first bounce as h1. The second bounce would then have a height of h2, the third bounce would have a height of h3, and so on. Since the heights form a geometric sequence, we can express this relationship as:
h2 = h1 * r
h3 = h2 * r = (h1 * r) * r = h1 * r^2
h4 = h3 * r = (h1 * r^2) * r = h1 * r^3
...
hn = h1 * r^(n-1)
In this case, the ball is dropped from a height of 500 meters, so h1 = 500. We need to find the height on the 8th bounce, so n = 8.
To determine the value of the common ratio (r), we can use the fact that the height on the first bounce is 500 meters and the height on the second bounce is given as 500 * r. Thus, we can set up the equation:
h2 = 500 * r
The table of values allows us to find the common ratio by dividing the height at the second bounce by the height at the first bounce:
h2 / h1 = (500 * r) / 500 = r
Looking at the table, we see that the height at the second bounce is 485 meters. Plugging in this value, we have:
r = 485 / 500
Using a calculator or performing the division, we find that r is approximately 0.97.
Now we can find the height on the 8th bounce by plugging in the values into the geometric sequence formula:
h8 = h1 * r^(8-1) = 500 * 0.97^7
Evaluating this expression, we find that h8 is approximately 376 meters.
Therefore, the ball bounces to a height of approximately 376 meters on the 8th bounce.