Final answer:
The total height of the first 10 bounces of a super ball thrown from the top of the Empire State Building can be calculated using the sum of a geometric series, considering that the ball bounces back 3/4 of its previous height each time, starting from 1460 feet.
Step-by-step explanation:
To find the total height of the first 10 bounces of a super ball thrown from the top of the Empire State Building, which is 1460 feet tall, we can use a geometric series. Since the ball bounces back up to 3/4 of its previous height with each bounce, we can express the total height traveled as a sum: initial descent + bounce up + second descent + bounce up + ..., and so on for 10 bounces. The initial descent is 1460 feet, and each subsequent bounce is 3/4 of the height of the previous fall or bounce.
The total height traveled can be calculated as follows:
- First bounce up: 1460 * (3/4)
- Second descent (same as first bounce up): 1460 * (3/4)
- Second bounce up: 1460 * (3/4)^2
- Third descent (same as second bounce up): 1460 * (3/4)^2
...and so on until the tenth bounce up.
The total height will then be the initial 1460 feet plus the sum of the heights of each subsequent bounce and fall, which essentially doubles the height traveled by the ball on each bounce (since the rise and fall are the equal for each bounce):
Total height = 1460 + 2 * (1460 * (3/4) + 1460 * (3/4)^2 + ... + 1460 * (3/4)^9).
This forms a geometric series, which can be summed using the formula for the sum of a geometric series:
S_n = a * (1 - r^n) / (1 - r)
Where:
a = first term in the series (1460),
r = common ratio between terms (3/4), and
n = number of terms (10).
The total height the ball will travel after the first 10 bounces can be found by plugging these values into the formula and adding the initial drop.