Question

A model rocket is launched with an initial upward velocity of 100 ft/s. The rocket's height h (in feet) after t seconds is given by the following.
h=100t-16t^2

Find all values of t for which the rocket's height is 34 feet.
Round your answer(s) to the nearest hundredth.

Answer 1

$\bf \stackrel{h}{34}=100t-16t^2\implies 16t^2-100t+34=0\\\\\\ 8t^2-50t+17=0 \\\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{8}t^2\stackrel{\stackrel{b}{\downarrow }}{-50}t\stackrel{\stackrel{c}{\downarrow }}{+17}=0 \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}$

$\bf t=\cfrac{ - (-50) \pm \sqrt { (-50)^2 -4(8)(17)}}{2(8)}\implies t=\cfrac{50\pm \sqrt{2500-544}}{16} \\\\\\ t=\cfrac{50\pm\sqrt{1956}}{16}\implies t=\cfrac{50\pm\sqrt{4\cdot 489}}{16}\implies t=\cfrac{50\pm\sqrt{2^2\cdot 489}}{16} \\\\\\ t=\cfrac{50\pm 2\sqrt{489}}{16}\implies t=\cfrac{25\pm \sqrt{489}}{8}\implies t\approx \begin{cases} 5.88916805\\ 0.36083195 \end{cases}$