A ball is dropped from a height of 500 meters. The table below shows the ball bounced 400 meters on the first bounce, 320 meters on the second bounce, and 256 meters on the third bounce. the heights form a geometric sequence. How high does the ball bounce on the 8th bounce? Round your answer to the nearest meter.
To solve this problem, we need to determine the common ratio in the geometric sequence. We can do this by examining the ratios between the heights of consecutive bounces.
The ratio between the second bounce and the first bounce is 320/400 = 0.8.
The ratio between the third bounce and the second bounce is 256/320 = 0.8.
Therefore, the common ratio is 0.8.
To find the height of the ball on the 8th bounce, we can multiply the height of the ball on the first bounce by the common ratio raised to the power of 7 (since we are counting the first bounce as the 1st bounce, not the 0th bounce).
Height of 8th bounce = 400 * (0.8)^7
Height of 8th bounce ≈ 400 * 0.2097152 ≈ 83.89 meters
Rounding to the nearest meter, the ball bounces approximately 84 meters on the 8th bounce.
To determine the height of the ball on the 8th bounce, we need to determine the common ratio of the geometric sequence.
The table shows that the ball bounces 400 meters on the first bounce, 320 meters on the second bounce, and 256 meters on the third bounce. Notice that each bounce is getting shorter by a factor of 0.8.
To find the common ratio, we take the ratio of any two consecutive terms:
Common ratio = 320 meters / 400 meters = 0.8
Now that we have the common ratio, we can find the height of the ball on the 8th bounce by multiplying the height of the third bounce (256 meters) by the common ratio (0.8) seven times:
Height of 8th bounce = 256 meters * (0.8)^7
Using a calculator or simplifying the expression, we find that the height of the 8th bounce is approximately 104.86 meters. Rounded to the nearest meter, the ball bounces approximately 105 meters on the 8th bounce.
To find out how high the ball bounces on the 8th bounce, we need to first determine the common ratio of the geometric sequence formed by the heights.
In a geometric sequence, each term is found by multiplying the previous term by a fixed number called the common ratio (r).
Let's calculate the common ratio:
Second bounce / First bounce = 320 / 400 = 0.8
Third bounce / Second bounce = 256 / 320 = 0.8
We can see that the ratios are equal, so the common ratio (r) is 0.8.
Now, we can use the formula to find the height of the ball on the 8th bounce:
Height on the 8th bounce = First bounce * (Common ratio)^(number of bounces - 1)
First bounce = 400
Common ratio (r) = 0.8
Number of bounces = 8
Height on the 8th bounce = 400 * (0.8)^(8-1)
Calculating the expression, we find:
Height on the 8th bounce = 400 * (0.8)^7 ≈ 400 * 0.2097152 ≈ 83.88608
Rounding this value to the nearest meter, the ball will bounce approximately 84 meters on the 8th bounce.