Question

write a quadratic function to model the vertical motion for each situation, given $h(t) = -16t^2 + v_0t + h_0$. find the maximum height. initial vertical velocity: 376 ft/s; initial height: 0 ft the function is $h(t) = \square$.

Explanation:

Step1: Substitute known values

Given $v_0=376$, $h_0=0$, substitute into $h(t)=-16t^2+v_0t+h_0$:
$h(t) = -16t^2 + 376t + 0 = -16t^2 + 376t$

Step2: Find time of max height

For quadratic $at^2+bt+c$, vertex time $t=-\frac{b}{2a}$. Here $a=-16$, $b=376$:
$t = -\frac{376}{2\times(-16)} = \frac{376}{32} = 11.75$

Step3: Calculate max height

Substitute $t=11.75$ into $h(t)$:
$h(11.75) = -16(11.75)^2 + 376(11.75)$
$= -16(138.0625) + 4418$
$= -2209 + 4418 = 2209$

Answer:

The function is $\boldsymbol{h(t) = -16t^2 + 376t}$, and the maximum height is $\boldsymbol{2209}$ feet.