Final answer:
The equation of the path is y = -(9/8)(x - 3)^2 + 18, which describes the parabolic trajectory of a football thrown and landing after 7 seconds with a peak height of 18 meters at 3 seconds.
Step-by-step explanation:
To find the equation of a parabolic path that a football follows, we can use the information provided about the football's highest point and the time it takes to land. First, we need to recognize that we are dealing with a quadratic equation in the form y = ax2 + bx + c, where y is the height and x is time. The coefficient a will be negative since the path is a downward opening parabola. The vertex of the parabola, which is the highest point, is at (3, 18) since it takes 3 seconds to reach the peak height of 18 meters. Finally, the football lands at y = 0 after 7 seconds, which gives us another point (7, 0) to help calculate the coefficients.
Given the vertex form of a parabola, y = a(x - h)2 + k, with vertex (h, k), we can substitute the vertex (3, 18) to get: y = a(x - 3)2 + 18.
To find the value of a, we use the fact that the football lands after 7 seconds at ground level (y=0). Substituting (7, 0) into the equation, we get: 0 = a(7 - 3)2 + 18.
Solving for a, we get: a = -18/16 = -9/8. Thus, the equation of the path is: y = -(9/8)(x - 3)2 + 18.