Question

A hamburger wrapper is thrown from a third floor dorm window, and its motion can be modeled by the equation h(t) = -3t^2 + 6t + 30, where h is the height in feet and t is the time in seconds.

a. Find the vertex of the parabola, then describe what it tells us physically about the situation.

b. Find the x-intercept, then describe what it tells us physically about the situation.

Answer 1

$h(t)=-3t^2+6t+30$

a. $a=-3,\ \ b=6,\ \ t=-\frac{b}{2a}=-\frac{6}{-6}=1$

$h(1)=-3+6+30=33$

Vertex of parabola: $t=1,\ \ h=33$ .

It means that the hamburger wrapper will reach the maximum height $h=33$ feet at the time $t=1$ second.

b. h(t)=0,\ \ $a=-3,\ \ b=6,\ \ c=30$

$t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}=\frac{-6\pm\sqrt{36+360}}{-6}=\frac{-6\pm 6\sqrt{11}}{-6}=1\pm\sqrt{11}$

$t_1=1-\sqrt{11}<0$ ($t$ is the time in seconds, so $t\geq 0$ )

$t_2=1+\sqrt{11}\approx 4.3$ seconds is the time when the hamburger wrapper reaches the ground (flight time)

Answers: a. $(1;\ 33)$ , b. $t=1+\sqrt{11}$ (and $1-\sqrt{11}$ )