A baseball is hit into the air and its height (h), in feet, above the ground after a time (t), in seconds, is modelled by the quadratic equation:
h(t) = –16t 2 + 96t + 3.
a) What is the maximum height of the ball?
b) How much time does it take to reach the maximum height?
max at t = 3
To find the maximum height and the time it takes to reach the maximum height for the given quadratic equation, we can use the vertex form of a quadratic equation. The vertex form is given by:
h(t) = a(t - t0)^2 + h0
where (t0, h0) is the vertex of the parabola.
a) The maximum height of the ball corresponds to the y-coordinate of the vertex. In this case, a = -16, so h(t) = -16t^2 + 96t + 3. To find the maximum height, we need to find the y-coordinate of the vertex.
The formula for the x-coordinate of the vertex (t-coordinate) is given by t = -b / (2a). In this case, b = 96 and a = -16. Substituting these values into the formula, we have:
t = -96 / (2 * (-16))
t = 96 / 32
t = 3
To find the y-coordinate of the vertex, we substitute the value of t = 3 into the equation h(t) = -16t^2 + 96t + 3:
h(3) = -16(3)^2 + 96(3) + 3
h(3) = -16(9) + 288 + 3
h(3) = -144 + 288 + 3
h(3) = 147
Therefore, the maximum height of the ball is 147 feet.
b) The time it takes to reach the maximum height is given by the x-coordinate of the vertex. In this case, the value of t is already calculated in part a) as t = 3.
Therefore, it takes 3 seconds for the ball to reach the maximum height.