Question

A ball is dropped from a height of 6 feet and begins bouncing.the height of each bounce is three-fourths the height of the previous bounce.find the total vertical distance travelled by the ball.

Answer 1

The total vertical distance traveled by the ball is 24 feet.

Step-by-step explanation:

The total vertical distance traveled by the ball can be calculated by adding up the distances of each bounce. Since each bounce is three-fourths the height of the previous bounce, we can write the equation for the total distance as:

Total distance = 6 + (3/4)*6 + (3/4)^2*6 + (3/4)^3*6 + ...

This is an infinite geometric series with a common ratio of 3/4. Using the formula for the sum of an infinite geometric series, we can calculate the total distance to be:

Total distance = 6 / (1 - 3/4) = 6 / (1/4) = 24 feet.

Answer 2

42 feet

Step-by-step explanation:

so when the ball hit the ground for the 1st time it travels a distance of 6 feet.

for later bounces let Di be the distance traveled up and down.

D1= 6 feet

D2 and D3 would be

`D2 =up+ down\\=6(3/4)+6(3/4)\\=12(3/4)`

similarly D3 is

`D3=6(3/4)(3/4)+6(3/4)(3/4)\\=12(3/4)^2`

by repeating this process we can find the vertical distance traveled by the ball

`D=6+12(3/4)+12(3/4)^2+......`

=6+12∑
`(3/4)^(n+1)`

where n=0,1,2,.....∞

=6+12(3/4)∑
`(3/4)^n`

`=6+9[1/(1-(3/4)]`

`=6+9(4)`

`=42 feet`