17. A rubber ball dropped on a hard surface takes a sequence of bounces, each one 3 / 5 as high as the preceding one. If this ball is dropped from a height of 10 feet, what is the total vertical distance it has traveled at the time when it hits the surface for its fifth bounce?
23 7/125 feet
36 14/125 feet
43 111/125 feet
46 14/125 feet
Make a sketch showing up to the fifth bounce and your
total distance = 10 + 2(10)(3/5) + 2(10)(3/5)^2 + 2(10)(3/5)^3 + 2(10)(3/5)^4
notice the first term does not fit the geometric series, so let's fix it
= [ 20(3/5) + 2(10)(3/5) + 2(10)(3/5)^2 + 2(10)(3/5)^3 + 2(10)(3/5)^4 ] - 10
a = 20
r = 3/5
n = 5
total distance = sum(5) - 10
= 20( 1 - (3/5)^5 )/(1-3/5)
= 20(5/2)(1 - 243/3125)
= 50(2882/3125)
= 5764/125 , I see that in your choices
To find the total vertical distance the rubber ball has traveled after its fifth bounce, we need to add up the distances of each bounce.
From the information given, we know that each bounce is 3/5 as high as the preceding one. If the first bounce is dropped from a height of 10 feet, we can calculate the height of each subsequent bounce as follows:
1st bounce: 10 feet
2nd bounce: (3/5) * 10 feet
3rd bounce: (3/5) * (3/5) * 10 feet
4th bounce: (3/5) * (3/5) * (3/5) * 10 feet
5th bounce: (3/5) * (3/5) * (3/5) * (3/5) * 10 feet
To simplify the calculation, we can express the fractions as decimals:
1st bounce: 10 feet
2nd bounce: (0.6) * 10 feet = 6 feet
3rd bounce: (0.6) * (0.6) * 10 feet = 3.6 feet
4th bounce: (0.6) * (0.6) * (0.6) * 10 feet = 2.16 feet
5th bounce: (0.6) * (0.6) * (0.6) * (0.6) * 10 feet = 1.296 feet
Now, we can add up these distances:
Total distance = 10 + 6 + 3.6 + 2.16 + 1.296 = 23.056 feet
Therefore, the total vertical distance the ball has traveled at the time when it hits the surface for its fifth bounce is approximately 23 feet.
To solve this problem, we need to determine the total vertical distance traveled by the rubber ball during its five bounces.
To begin, let's calculate the height of each bounce. Given that each bounce is 3/5 as high as the preceding one, we can set up the following sequence:
Bounce 1: 10 feet
Bounce 2: (3/5) * 10 feet
Bounce 3: (3/5) * (3/5) * 10 feet
Bounce 4: (3/5) * (3/5) * (3/5) * 10 feet
Bounce 5: (3/5) * (3/5) * (3/5) * (3/5) * 10 feet
Now, let's simplify each term:
Bounce 1: 10 feet
Bounce 2: 6 feet
Bounce 3: (3/5) * 6 feet = 18/5 feet
Bounce 4: (3/5) * (18/5) feet = 54/25 feet
Bounce 5: (3/5) * (54/25) feet = 162/125 feet
Finally, to find the total vertical distance traveled by the ball during its five bounces, we add up all the heights:
Total distance = 10 + 6 + 18/5 + 54/25 + 162/125
= 1250/125 + 750/125 + 18/5 + 54/25 + 162/125
= (1250 + 750 + 18 + 54 + 162)/125
= 2234/125
Therefore, the total vertical distance traveled by the rubber ball at the time of its fifth bounce is 2234/125 feet. However, none of the provided answer choices match this result.