Question

Derek hits a baseball thrown by the pitcher with an initial vertical velocity of 60 ft/sec from a height of 3 feet.

a. What type of function models the height of the baseball versus time since it was hit?
b. Write a function that models the height, h (in feet), the baseball travels over a period of time in t seconds.
c. if no one touches the ball, how long is it in the air? Justify your answer mathematically.

Answer 1

Check the picture below.

`~~~~~~\textit{initial velocity in feet} \\\\ h(t) = -16t^2+v_ot+h_o \quad \begin{cases} v_o=\textit{initial velocity}&60\\ \qquad \textit{of the object}\\ h_o=\textit{initial height}&3\\ \qquad \textit{of the object}\\ h=\textit{object's height}&\\ \qquad \textit{at`

how long is the ball in the air? Well, let's find how many "t" seconds went by, by the time it hit the ground, that is, by the time h(t) = 0.

`\stackrel{h(t)}{0}=-16t^2+60t+3\implies 16t^2-60t-3=0 \\\\[-0.35em] ~\dotfill\\\\ ~~~~~~~~~~~~\textit{quadratic formula} \\\\ \stackrel{\stackrel{a}{\downarrow }}{16}t^2\stackrel{\stackrel{b}{\downarrow }}{-60}t\stackrel{\stackrel{c}{\downarrow }}{-3}=0 \qquad \qquad t= \cfrac{ - b \pm \sqrt { b^2 -4 a c}}{2 a}`

`t= \cfrac{ - (-60) \pm \sqrt { (-60)^2 -4(16)(-3)}}{2(16)} \implies t = \cfrac{ 60 \pm \sqrt { 3600 +192}}{ 32 } \\\\\\ t= \cfrac{ 60 \pm \sqrt { 3792 }}{ 32 }\implies t= \cfrac{ 60 \pm 4\sqrt { 237 }}{ 32 }\implies t=\cfrac{ 15 \pm \sqrt { 237 }}{ 8 } \\\\\\ t= \begin{cases} \frac{ 15 + \sqrt { 237 }}{ 8 }\\\\ \frac{ 15 - \sqrt { 237 }}{ 8 } \end{cases}\implies t\approx \begin{cases} ~~ ~3.799 ~~ \checkmark \qquad \textit{about 4 seconds}\\\\ -0.049 ~~ \bigotimes \end{cases}`

since the seconds can't be negative for specific case, so we toss that negative value.