Question

a rocket is launched in the air. its height in feet is given by h(t)=-16t² + 120t where t represents the time in seconds after launch. interpret the coordinates of the vertex in context. answer attempt 1 out of 2 the x - coordinate (or t - coordinate) of the vertex is and represents the y - coordinate (or h - coordinate) of the vertex is and represents

Explanation:

Step1: Recall vertex - formula for a quadratic function

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 $h(t)=-16t^{2}+120t$, where $a=-16$ and $b = 120$.

Step2: Calculate the $t$ - coordinate of the vertex

$t=-\frac{120}{2\times(-16)}=\frac{120}{32}=\frac{15}{4} = 3.75$ seconds. This represents the time when the rocket reaches its maximum height.

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

Substitute $t = 3.75$ into the function $h(t)=-16t^{2}+120t$. So $h(3.75)=-16\times(3.75)^{2}+120\times3.75=-16\times14.0625 + 450=-225+450 = 225$ feet. This represents the maximum height of the rocket.

Answer:

The x - coordinate (or t - coordinate) of the vertex is $3.75$ and represents the time in seconds when the rocket reaches its maximum height.
The y - coordinate (or h - coordinate) of the vertex is $225$ and represents the maximum height of the rocket in feet.