A ball rebounds 7/8ths as high as it bounced on the previous bounce and is dropped from a height of 8 feet. How high does it bounce on the fourth bounce and how far has it traveled after the fourth bounce?
Height after 4th bounce = 8x(7/8)^4
Total = 8 + 8*(7/8)*2 + 8*(7/8)^2*2 + 8*(7/8)^3*2 + 8*(7/8)^4
To solve this problem, we'll use the concept of geometric sequences since the ball rebounds a fraction of its previous height. Let's break down the problem step by step:
Step 1: Understanding the given information
We know that the ball rebounds 7/8ths as high as it bounced on the previous bounce. This means that each consecutive bounce will be 7/8 times the height of the previous bounce.
We also know that the ball is dropped from a height of 8 feet. This information will help us calculate the height of each bounce.
Step 2: Calculating the height of each bounce
Since the first bounce is from a height of 8 feet, let's denote it as h₁. The height of the second bounce, h₂, can be calculated as 7/8 times h₁. Subsequently, the height of the third bounce, h₃, can be calculated as 7/8 times h₂, and so on.
Using this pattern, we can calculate the height of each subsequent bounce. Since we need to find the height of the fourth bounce, let's calculate h₄:
h₁ = 8 feet (given)
h₂ = (7/8) * h₁
h₃ = (7/8) * h₂
h₄ = (7/8) * h₃
Step 3: Calculating the height of the fourth bounce
Now, let's substitute the values to find the height of the fourth bounce, h₄:
h₁ = 8 feet
h₂ = (7/8) * 8 feet
h₃ = (7/8) * [(7/8) * 8 feet]
h₄ = (7/8) * [(7/8) * [(7/8) * 8 feet]]
By simplifying the expression, we can find the value of h₄, which represents the height of the fourth bounce.
Step 4: Calculating the distance traveled after the fourth bounce
To find the total distance traveled after the fourth bounce, we need to sum up the distances covered in each individual bounce.
Since the distance covered in each bounce is the same as the height, the total distance traveled after the fourth bounce is the sum of the first four bounces:
Total Distance = h₁ + h₂ + h₃ + h₄
By substituting the values we've calculated for each bounce, we can find the total distance traveled by the ball after the fourth bounce.
Now, I will calculate the value of h₄ and the total distance traveled after the fourth bounce based on the information provided.