33 an object is launched upward at 64 feet per second from a platform 80 feet above the ground. the function…

33 an object is launched upward at 64 feet per second from a platform 80 feet above the ground. the function s(t) models the height of the object t seconds after launch. if s(t)=-16t² + 64t + 80, state the vertex of s(t), and explain in detail what each coordinate means in the context of the problem. after the object is launched, how many seconds does it take for the object to hit the ground? justify your answer.

33 an object is launched upward at 64 feet per second from a platform 80 feet above the ground. the function s(t) models the height of the object t seconds after launch. if s(t)=-16t² + 64t + 80, state the vertex of s(t), and explain in detail what each coordinate means in the context of the problem. after the object is launched, how many seconds does it take for the object to hit the ground? justify your answer.

Answer

Explanation:

Step1: Find the x - coordinate of the vertex

For a quadratic function $y = ax^{2}+bx + c$, the x - coordinate of the vertex is given by $t=-\frac{b}{2a}$. In the function $s(t)=-16t^{2}+64t + 80$, $a=-16$ and $b = 64$. So, $t=-\frac{64}{2\times(-16)}=\frac{-64}{-32}=2$.

Step2: Find the y - coordinate of the vertex

Substitute $t = 2$ into the function $s(t)$. $s(2)=-16\times2^{2}+64\times2 + 80=-16\times4+128 + 80=-64+128 + 80=144$. The vertex is $(2,144)$. The x - coordinate $t = 2$ means that 2 seconds after the object is launched, it reaches its maximum height. The y - coordinate $s(2)=144$ means that the maximum height of the object is 144 feet.

Step3: Find when the object hits the ground

Set $s(t)=0$, so $-16t^{2}+64t + 80 = 0$. Divide the entire equation by -16 to simplify: $t^{2}-4t - 5=0$. Factor the quadratic equation: $(t - 5)(t+1)=0$. Set each factor equal to zero: $t - 5=0$ gives $t = 5$ and $t+1=0$ gives $t=-1$. Since time $t\geq0$ in this context, we discard $t=-1$.

Answer:

The vertex of $s(t)$ is $(2,144)$. The x - coordinate 2 means the time (in seconds) when the object reaches its maximum height, and the y - coordinate 144 means the maximum height (in feet) of the object. It takes 5 seconds for the object to hit the ground.