An archer shoots an arrow into the air such that its height at any time, t, is given by the function h(t) = -16t^2 + kt + 3. If the maximum height of the arrow occurs at time t = 4, what is the value of k?
(1) 128 (3) 8
(2) 64 (4) 4
An archer shoots an arrow into the air such that its height at any time, t, is given by the function h(t) = -16t^2 + kt + 3. If the maximum height of the arrow occurs at time t = 4, what is the value of k?
(1) 128 (3) 8
(2) 64 (4) 4
All right. In that case we need to look at the height function a bit more carefully. It is an inverted parabola. The maximum height will be at the vertex of the parabola, which is on the axis of symmetry.
Given a parabola $\displaystyle y = ax^2 + bx + c$ the axis of symmetry will be the line $\displaystyle x = -\frac{b}{2a}$.
We have the parabola $\displaystyle h = -16t^2 + kt + 3$. We know that the location of the max height is the line t = 4, which is our axis of symmetry. So:
$\displaystyle t = - \frac{k}{2 \cdot -16} = 4$
So
$\displaystyle k = 128$.
-Dan