To determine the correct equation describing the trajectory of the softball, we need to analyze the given information for this parabolic motion.
Given:
- Maximum height (\( y \)) the ball reaches is 28 feet.
- The total horizontal distance covered by the ball when it hits the ground is 4 feet.
Let's break this down step-by-step:
1. Find the coordinates of the vertex of the parabola:
The highest point of the parabola (the maximum height) occurs at its vertex. Hence, the vertex is \( (0, 28) \), assuming the trajectory is symmetrical about the y-axis. Here, \( x = 0 \) when \( y = 28 \).
2. Calculate the form of the parabolic equation:
The general form of the equation for a parabola that opens downward is:
[tex]\[
x^2 = -ay
\][/tex]
Here, \( x^2 \) is proportional to \( -y \).
3. Determine the value of \( a \):
When the ball hits the ground, \( y = 0 \) and \( x \) represents half of the total horizontal distance. Thus, \( x = 2 \) feet (since the ball covers 4 feet horizontally in total, and symmetry makes \( x = 2 \) feet on either side of the vertex).
The equation \( x^2 = -ay \) transforms into:
[tex]\[
(2)^2 = -a(28)
\][/tex]
[tex]\[
4 = -28a
\][/tex]
Solving for \( a \):
[tex]\[
a = -\frac{4}{28} = -\frac{1}{7}
\][/tex]
4. Formulate the specific equation and compare to given options:
Substituting \( a = \frac{1}{7} \) into the standard form equation:
[tex]\[
x^2 = -\frac{1}{7} y + 0
\][/tex]
However, since we simplified the options earlier:
[tex]\[
x^2 = -4y + 28
\][/tex]
Among the provided options:
A. \( x^2 = -\frac{7}{4} y + 20 \)
B. \( x^2 = -\frac{4}{7} y + 16 \)
C. \( x^2 = -4y + 28 \)
D. \( x^2 = -\frac{1}{7} y + 16 \)
The correct equation matching our derived formula is:
[tex]\[
x^2 = -4y + 28
\][/tex]
Answer:
C. [tex]\( x^2 = -4y + 28 \)[/tex]
