Question

A ball is dropped from a height of 20 feet. Each time it bounces, it reaches ⅓ of its previous height. What is the total vertical distance the ball has traveled when it hits the ground the fourth time?

Answer 1

The total vertical distance the ball has traveled when it hits the ground the fourth time is approximately 141.48 feet.

Step-by-step explanation:

To find the total vertical distance traveled by the ball when it hits the ground the fourth time, we need to find the sum of the distances it travels each time it bounces. We know that each bounce reaches 1/3 of the previous height. Let's calculate the distances:

First bounce: 20 feet

Second bounce: (1/3) * 20 feet

Third bounce: (1/3) * ((1/3) * 20 feet)

Fourth bounce: (1/3) * ((1/3) * ((1/3) * 20 feet))

To find the total distance, we add up all the distances:

Total distance = 20 + (1/3) * 20 + (1/3) * ((1/3) * 20) + (1/3) * ((1/3) * ((1/3) * 20))

Simplifying the expression, we get:

Total distance = 20 + (1/3) * 20 + (1/3)^2 * 20 + (1/3)^3 * 20

Total distance = 20 + 20/3 + 20/9 + 20/27

Total distance = 3820/27 = 141.48 feet

Answer 2

You have to calculate the sum of 20 + 20/3 + 20/ 9 + 20/27 + 20/ 81 + ...

That is a geometric series with first term 20 and r = 1/3.

Now you can use the formula for the sum of a geometric series:

∑ Ai from i = 0 to i = A / ( 1 - r), where A is the first term.

=> 20 + 20/3 + 20/ 9 + 20/27 + 20/ 81 + ... = 20 / ( 1 - 1/3) = 20 / (2/3) = 20*3/2 = 30.

Answer: 30 feet.