Answer: h(t) = -2(t - 3)^2 + 38
Explanation:
To find the equation of the quadratic function that models the height of the ball h(t) at time t, we need to use the vertex form of a quadratic equation.
The vertex form of a quadratic equation is given by:
h(t) = a(t - h)^2 + k where (h, k) is the vertex of the parabola.
Given the information:
After 1 second, the ball is 30 feet high. This gives us the point (1, 30). After 3 seconds, the ball reaches its maximum height of 38 feet. This gives us the vertex (3, 38). After 5 seconds, it returns to a height of 30 feet. This gives us the point (5, 30).
We can use these three points to find the equation of the quadratic function.
1. Find the value of a: Using the vertex form, we substitute the vertex point (3, 38) into the equation: 38 = a(3 - 3)^2 + k 38 = a(0)^2 + k 38 = k
2. Substitute the vertex point and another point into the equation:
Using the point (1, 30), we substitute it into the equation: 30 = a(1 - 3)^2 + 38 30 = a(-2)^2 + 38 30 = 4a + 38 4a = -8 a = -2 3.
Substitute the value of a and the vertex point into the equation: The equation becomes: h(t) = -2(t - 3)^2 + 38