Final answer:
The total height of the bounces of a super ball thrown from the Empire State Building until it stops bouncing is 4380 feet, calculated by subtracting the initial drop from the sum of an infinite geometric series.
Step-by-step explanation:
The question is asking about the total distance a super ball travels in its bounces after being thrown off the top of the Empire State Building, which is 1460 feet tall. The ball bounces back up 3/4 of the distance it falls each time. To find the total distance, we can calculate the sum of an infinite geometric series where the first term is the height of the building (1460 feet) and the common ratio is 3/4.
To calculate the sum of this geometric series, we can use the formula: Sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. Substituting the values, we get: Sum = 1460 / (1 - 3/4) = 1460 / (1/4) = 1460 * 4 = 5840 feet. This includes the initial drop and all subsequent bounces.
However, we must subtract the initial drop because the question only asks for the total height of the bounces. Therefore, the sum of the distances of all the bounces (excluding the first fall) is 5840 feet - 1460 feet which equals 4380 feet. This value represents the infinite sum of the bounce heights until the ball stops bouncing.