Question

A catapult launches a boulder with an upward velocity of 92 m/s. the height of the boulder, h, in meters after t seconds is given by the function h = -5t^2 + 92t + 16. how long does it take to reach maximum height? what is the boulder's maximum height? Round to the nearest hundredth, if necessary.

Answer 1

Check the picture below.

$\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ h=\stackrel{\stackrel{a}{\downarrow }}{-5}t^2\stackrel{\stackrel{b}{\downarrow }}{+92}t\stackrel{\stackrel{c}{\downarrow }}{+16} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)$

$\bf \left( -\cfrac{92}{2(-5)}~~,~~16-\cfrac{92^2}{4(-5)} \right)\implies \left(\cfrac{46}{5}~~,~~16+\cfrac{2116}{5} \right) \\\\\\ \left(\cfrac{46}{5}~~,~~\cfrac{2196}{5} \right)\implies \left(\stackrel{\textit{how long it took}}{9\frac{1}{5}}~~,~~\stackrel{\textit{how high it went}}{439\frac{1}{5}} \right)$