Question

fireworks are shot by remote control into the air from a pit that is 9 ft below earths surface. (a) find an equation that models the height of a roman candle t seconds after it is shot upward with an initial velocity of 80 ft/sec. graph the equation. (b) what is the maximum height above ground - level that the roman candle will reach? how many seconds will it take to reach that height? (a) find an equation that models the height of the roman candle (in feet) t seconds after it is shot upward with an initial velocity of 80 ft/sec. s(t)=-16t^{2}+80t - 9 (simplify your answer.) graph the equation. choose the correct graph below. a. 0,5 by -9,108 b. 0,5 by -9,108 c. 0,5 by -9,108 d. 0,5 by -9,108

Explanation:

Step1: Recall the motion - equation

The general equation for vertical - motion under gravity is $s(t)=-16t^{2}+v_{0}t + s_{0}$, where $v_{0}$ is the initial velocity and $s_{0}$ is the initial position. Here, $v_{0} = 80$ ft/sec and $s_{0}=-9$ ft. So, $s(t)=-16t^{2}+80t - 9$.

Step2: Analyze the graph of the quadratic function

The function $s(t)=-16t^{2}+80t - 9$ is a quadratic function of the form $y = ax^{2}+bx + c$ with $a=-16$, $b = 80$, and $c=-9$. Since $a=-16<0$, the parabola opens downwards. The vertex of the parabola $y = ax^{2}+bx + c$ has $t$ - coordinate $t=-\frac{b}{2a}$.

Step3: Calculate the $t$ - coordinate of the vertex

$t=-\frac{80}{2\times(-16)}=\frac{80}{32}=\frac{5}{2}=2.5$ seconds.

Step4: Calculate the maximum height

Substitute $t = 2.5$ into $s(t)$: $s(2.5)=-16\times(2.5)^{2}+80\times2.5 - 9=-16\times6.25+200 - 9=-100 + 200 - 9=91$ ft.

The graph of the quadratic function $s(t)=-16t^{2}+80t - 9$ is a parabola opening downwards. All the given graphs have the same viewing - window $[0,5]$ by $[-9,108]$. Since the parabola opens downwards, we need to find the graph that has a vertex at $t = 2.5$ seconds and maximum value around $s = 91$ ft.

Answer:

The equation is $s(t)=-16t^{2}+80t - 9$. The maximum height above ground level is 91 ft and it takes 2.5 seconds to reach that height. Without seeing the actual differences in the graphs (since they are not described in detail other than the viewing - window), we can't definitively choose the correct graph from A, B, C, and D. But the graph should be a parabola opening downwards with vertex at $(2.5,91)$ within the viewing - window $[0,5]$ by $[-9,108]$.