A ball is placed on a long inclined ramp. It rolls down the ramp as shown in the diagram.
The equation d = 23 · t2 represents the distance in feet, d, the ball travels in t seconds.
Part A
How far does the ball travel in 5 seconds? Show your work.
Part B
How many seconds would it take for the ball to travel 25 feet? Show your work.
Part C
If the ball traveled for 9 seconds, what would be the distance it traveled? Show your work.
Part D
In the inequality shown below, x and y are consecutive, positive integers. X<T<Y
If T is the time it takes for the ball to travel 1,000 feet, what are the values of x and y? Explain your answer.
Part A:
To find how far the ball travels in 5 seconds, we can plug in t = 5 into the equation d = 23 · t^2.
d = 23 · (5)^2
d = 23 · 25
d = 575
Therefore, the ball travels 575 feet in 5 seconds.
Part B:
To find how many seconds it would take for the ball to travel 25 feet, we need to solve the equation 25 = 23 · t^2 for t.
25 = 23 · t^2
t^2 = 25/23
t^2 ≈ 1.08696
t ≈ √1.08696
t ≈ 1.042
Therefore, it would take approximately 1.042 seconds for the ball to travel 25 feet.
Part C:
To find the distance the ball would travel in 9 seconds, we can plug in t = 9 into the equation d = 23 · t^2.
d = 23 · (9)^2
d = 23 · 81
d = 1863
Therefore, the ball would travel 1863 feet in 9 seconds.
Part D:
To find the values of x and y, we need to determine the consecutive positive integers between which 1,000 feet falls.
Let's use the equation d = 23 · t^2 and solve for t:
1,000 = 23 · t^2
t^2 = 1,000/23
t^2 ≈ 43.4783
t ≈ √43.4783
t ≈ 6.595
Since x and y are consecutive positive integers, we can conclude that x = 6 and y = 7.
Therefore, the values of x and y in the inequality are x = 6 and y = 7.
Part A: To find how far the ball travels in 5 seconds, we can substitute t = 5 into the equation d = 23 · t^2:
d = 23 · (5)^2
d = 23 · 25
d = 575 feet
Therefore, the ball travels 575 feet in 5 seconds.
Part B: To find how many seconds it would take for the ball to travel 25 feet, we can rearrange the equation d = 23 · t^2 to solve for t:
d = 23 · t^2
25 = 23 · t^2
Divide both sides by 23:
25/23 = t^2
Taking the square root of both sides:
sqrt(25/23) = t
Approximately:
t ≈ 1.04 seconds
Therefore, it would take approximately 1.04 seconds for the ball to travel 25 feet.
Part C: To find the distance the ball traveled in 9 seconds, we can substitute t = 9 into the equation d = 23 · t^2:
d = 23 · (9)^2
d = 23 · 81
d = 1863 feet
Therefore, the ball would travel 1863 feet in 9 seconds.
Part D: In this part, we know that x and y are consecutive, positive integers and T represents the time it takes for the ball to travel 1000 feet.
From our equation, d = 23 · t^2, we can set up the following inequality to solve for T:
1000 ≤ 23 · T^2
Divide both sides by 23:
1000/23 ≤ T^2
Taking the square root of both sides:
sqrt(1000/23) ≤ T
Approximately:
T ≥ 7.39 seconds
Since T represents the time it takes for the ball to travel 1000 feet, it cannot be less than 7.39 seconds. Therefore, x would be 7 and y would be 8, because they are consecutive positive integers.