To choose the correct equation to model the distance of the rock over time, we need to consider the general context and behavior of objects thrown into the air.
When an object is thrown from a height (in this case, from a 15-foot bridge), its motion can typically be modeled by a quadratic equation, which has the form:
\[ y = ax^2 + bx + c \]
where:
- \( y \) represents the height of the rock over time,
- \( a \) is a negative value indicating the object is subject to gravity (i.e., it will eventually fall),
- \( b \) and \( c \) are coefficients that affect the shape and position of the parabola.
Given the provided options, there are two equations with a quadratic term:
- y = -2.4x² + 5.6x + 14.9
- y = -1.5x² + 4.6x + 14.7
- y = -1.5x² + 4.6x + 14.8
To determine the most appropriate equation, consider:
- The initial height of the bridge (15 feet) should be reflected in the constant term (\( c \)).
- As time progresses (\( x \) increases), the distance \( y \) should decrease since the rock is falling.
Since the bridge height should inform our constant term, we would expect \( c \) to be close to 15. Among the options:
- The first equation has a constant term of 14.9, which is close to 15.
- The second equation has a constant term of 14.7.
- The third equation has a constant term of 14.8.
All of these are reasonable approximations for an initial height of 15 feet, but the first one is the furthest from 15.
Without additional details on the projectile's initial velocity or the behavior of the distances over time, it's hard to determine the exact answer.
If we were to choose based on the information available, I'd suggest going with:
y = -1.5x² + 4.6x + 14.8
This keeps a close approximation to the expected initial height while being the most rounded in terms of coefficients.