A ball has a 75% rebound ratio. When you drop it from a height of 16 ft, it bounces and bounces...
When it strikes the ground for the second time, the ball has traveled a total of 28 ft in a downward direction. How far downward has the ball traveled when it strikes the ground for the 17th time?
T(1)=16
T(2)=12
T(3)=9
...
T(n)=16(3/4)^(n-1)
It is a geometric progression (GP) with a ratio of 3/4.
The sum to n terms of a GP is:
S(n)=16(1-(3/4)^n)/(1-(3/4))
Substitute n=17 in the above formula to get S(17), the total downward distance.
Well, it seems like this ball has quite the bouncy personality! With a 75% rebound ratio, it keeps bouncing back for more. Now, let's calculate how far downward the ball has traveled when it strikes the ground for the 17th time.
We know that when the ball strikes the ground for the second time, it has traveled a total of 28 ft downward. Since each bounce decreases the distance traveled by 25%, we can calculate the distance traveled after each bounce.
After the second bounce:
28 ft * 0.75 = 21 ft
After the third bounce:
21 ft * 0.75 = 15.75 ft
After the fourth bounce:
15.75 ft * 0.75 = 11.8125 ft
And so on...
By the time we get to the 17th bounce, we'll have traveled approximately:
1.785 ft (rounded to 3 decimal places)
So, when the ball strikes the ground for the 17th time, it will have traveled approximately 1.785 ft downward. Keep bouncing, ball!
To find the distance the ball has traveled when it strikes the ground for the 17th time, we need to determine the total distance traveled during each bounce:
1st bounce: 16 ft downward
2nd bounce: (16 ft * 0.75) + 16 ft = 12 ft + 16 ft = 28 ft downward
From the given information, we can see that after the 2nd bounce, the total distance traveled downward is 28 ft. Thus, we can conclude that each bounce covers a distance of 28 ft.
Therefore, the distance traveled when the ball strikes the ground for the 17th time would be:
Distance = Total distance covered in each bounce * (Number of bounces - 1)
= 28 ft * (17 - 1)
= 28 ft * 16
= 448 ft downward
Thus, the ball has traveled 448 ft downward when it strikes the ground for the 17th time.
To find out how far downward the ball has traveled when it strikes the ground for the 17th time, we need to determine the height of each bounce.
The ball has a rebound ratio of 75%, which means it bounces back up to 75% of its previous height after each bounce.
When the ball is dropped from a height of 16 ft, it first bounces back up to (75/100) * 16 ft = 12 ft.
Since the total distance traveled when the ball strikes the ground for the second time is 28 ft, we can calculate the height of the second bounce by subtracting the initial drop distance from the total distance traveled.
Second bounce height = Total distance - Initial drop distance
Second bounce height = 28 ft - 16 ft
Second bounce height = 12 ft
Now, to find the height of the subsequent bounces, we need to calculate 75% of the previous bounce height.
Third bounce height = (75/100) * 12 ft = 9 ft
Fourth bounce height = (75/100) * 9 ft = 6.75 ft
Fifth bounce height = (75/100) * 6.75 ft = 5.0625 ft
This pattern continues until we have calculated the height of the 17th bounce.
To sum up the total distance traveled when the ball strikes the ground for the 17th time, we need to add up the initial drop distance and the distance covered in each bounce.
Total distance = Initial drop distance + Sum of bounce heights
Total distance = 16 ft + 12 ft + 9 ft + 6.75 ft + 5.0625 ft + ...
Calculating the sum of bounce heights can become repetitive. However, we can use a geometric series formula to simplify this calculation.
The formula for the sum of a geometric series is given as:
S = a * (1 - r^n) / (1 - r)
Where:
S = sum of the series
a = first term of the series
r = common ratio between terms
n = number of terms in the series
In this case, a = 12 ft (the first bounce height) and r = 75% or 0.75 (the common ratio). We want to find the sum of the first 17 terms (n = 17).
Substituting these values into the formula, we get:
Total distance = 16 ft + 12 ft * (1 - 0.75^17) / (1 - 0.75)
Using a calculator, we can evaluate this expression to find the total distance traveled when the ball strikes the ground for the 17th time.