To determine the ball's height above the ground when its horizontal distance from the robot is 0 inches, we need to develop a parabolic model that best fits the given data points.
Here are the steps we followed:
1. Plot the Given Data Points:
We have the following data points for horizontal distance ([tex]\(x\)[/tex]) and height ([tex]\(y\)[/tex]):
- (8, 42)
- (10, 32)
- (12, 18)
- (14, 0)
2. Assume a Parabolic Equation:
Since the trajectory of the ball is likely a parabola, we assume a quadratic relationship of the form:
[tex]\[ y = ax^2 + bx + c \][/tex]
3. Determine the Coefficients:
Using the given data points, we fit a second-degree polynomial to the data. The coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] obtained from the fit are:
[tex]\[ a = -0.5 \][/tex]
[tex]\[ b = 4 \][/tex]
[tex]\[ c = 42 \][/tex]
4. Calculate the Height at Horizontal Distance 0:
We now substitute [tex]\(x = 0\)[/tex] into our quadratic equation:
[tex]\[ y = -0.5(0)^2 + 4(0) + 42 \][/tex]
[tex]\[ y = 42 \][/tex]
Therefore, the ball's height above the ground when its horizontal distance from the robot is 0 inches is [tex]\(42\)[/tex] inches.
The coefficients of the fitted polynomial confirm that our calculations are correct, and the height at [tex]\(x = 0\)[/tex] is indeed [tex]\(42\)[/tex] inches, very close to [tex]\(42\)[/tex]. The very slight difference of [tex]\(41.9999999999999\)[/tex] is due to numerical approximations. Thus, you can safely conclude that the ball's height is approximately [tex]\(42\)[/tex] inches, considering reasonable numerical tolerance.
