POSSIBL A student rolls a marble down a 5 ft ramp several different
times. She records the time it takes for the marble to roll
several various distances in the table below.
1
d
(sec) (meters)
0.3 0.07
0.6
0.27
0.91
1.1
1.5
1.69
w
2.1
3.31
2.4
4.32
About how far up the ramp should she release the ball if
she wants it to reach the bottom in 1.9 seconds?

The distance up the ramp in meters she should release the ball if she wants it to reach the bottom in 1.9 seconds is approximately 10 meters

Step-by-step explanation:

The given parameters of the motion of Kathy are expressed in a tabular form after sorting in increasing order using Microsoft Excel, as follows, as follows;

The general form of the quadratic equation is d = a·t² + b·t + c

Taking the points (0.07, 1.5), (0.27, 1.69), and (0.6, 2.4), which are shown to best follow the path of a quadratic (parabolic) curve by using the Chart function to plot the data on Microsoft Excel, we have;

1.5 = a·0.07² + b·0.07 + c = 0.0049·a + 0.07·b + c

1.69 = a·0.27² + b·0.27 + c = 0.0729·a + 0.27·b + c

2.4 = a·0.6² + b·0.6 + c = 0.36·a + 0.6·b + c

1.5 = 0.0049·a + 0.07·b + c

1.69 = 0.0729·a + 0.27·b + c

2.4 = 0.36·a + 0.6·b + c

Solving with the above system of equations, we have;

a ≈ 2.267

b ≈ 0.179

c ≈ 1.476

Therefore, we have;

d ≈ 2.267·t² + 0.179·t + 1.476

The distance, 'd', up the ramp in meters from which she should release the ball so that the ball reaches the bottom in t = 1.9 seconds is given as follows;

d ≈ 2.267 × (1.9)² + 0.179 × (1.9) + 1.476 = 9.9997

The distance up the ramp in meters she should release the ball if she wants it to reach the bottom in t = 1.9 seconds is d ≈ 9.9997 meters which is approximately 10 meters.