Question #05734

Aug 21, 2016

$5$ feet

Explanation:

Take a look at the graph of this equation.
graph{-16x^2+8x+4 [-10, 10, -5, 5]}

We can see that the maximum height of the object occurs at the vertex of the parabola. In order to find the time ${t}_{\max}$ where the object reaches the vertex, we can use the following formula:
${t}_{\max} = - \frac{b}{2 a}$

This is the formula used throughout algebra to find the $x$-coordinate of the vertex of a parabola.

In the equation $- 16 {t}^{2} + 8 t + 4$, we have $a = - 16$ and $b = 8$, so:
${t}_{\max} = - \frac{8}{2 \left(- 16\right)} = \frac{1}{4}$ seconds

This tells us that the maximum height of the object is reached at $t = \frac{1}{4}$ seconds. However, the question asks for the actual height, not the time. To find the height, we simply plug in $\frac{1}{4}$ for $t$:
$S \left(t\right) = - 16 {t}^{2} + 8 t + 4$
${S}_{\max} \left(t\right) = S \left({t}_{\max}\right) = S \left(\frac{1}{4}\right) = - 16 {\left(\frac{1}{4}\right)}^{2} + 8 \left(\frac{1}{4}\right) + 4$
$= 5$ feet