Question

a tennis ball machine serves a ball vertically into the air from a height of 2 feet, with an inital speed of 120 feet per second. What is the maximum height, in feet, the ball will attain?

Answer 1

Check the picture below.

$\bf ~~~~~~\textit{initial velocity}\\\\ \begin{array}{llll} ~~~~~~\textit{in feet}\\\\ h(t) = -16t^2+v_ot+h_o \\\\ \end{array} \quad \begin{cases} v_o=\stackrel{\textit{initial velocity of the object}}{120}\\\\ h_o=\stackrel{\textit{initial height of the object}}{2}\\\\ h=\stackrel{}{\textit{height of the object at "t" seconds}} \end{cases} \\\\\\ h(t)=-16t^2+120t+2$

 $\bf \textit{ vertex of a vertical parabola, using coefficients}\\\\ \begin{array}{llll} h(t)= &{{ -16}}t^2&{{ +120}}t&{{ +2}}\\ &\uparrow &\uparrow &\uparrow \\ &a&b&c \end{array}\qquad \left(-\cfrac{{{ b}}}{2{{ a}}}\quad ,\quad {{ c}}-\cfrac{{{ b}}^2}{4{{ a}}}\right) \\\\\\ \left(-\cfrac{120}{2(-16)}~~,~~2-\cfrac{120^2}{4(-16)} \right)\implies \left( \cfrac{15}{4}~~,~~2+\cfrac{14400}{32} \right)$

$\bf \left( \cfrac{15}{4}~~,~~2+450 \right)\implies \left( \stackrel{seconds}{3\frac{3}{4}}~~,~~\stackrel{feet}{452} \right)$