Question

question
a rocket is launched in the air. its height in feet is given by $h = -16t^2 + 64t$ 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 \boxed{} and represents
\boxed{} \boxed{}.
the y - coordinate (or h - coordinate) of the vertex is \boxed{} and represents
\boxed{} \boxed{}.

Explanation:

Step1: Find t-coordinate of vertex

For a quadratic function \( h = at^2 + bt + c \), the t - coordinate of the vertex is given by \( t=-\frac{b}{2a} \). In the equation \( h=-16t^{2}+64t \), \( a = - 16 \) and \( b = 64 \). So, \( t=-\frac{64}{2\times(-16)}=-\frac{64}{-32} = 2 \).

Step2: Find h-coordinate of vertex

Substitute \( t = 2 \) into the equation \( h=-16t^{2}+64t \). \( h=-16\times(2)^{2}+64\times2=-16\times4 + 128=-64 + 128 = 64 \).

Step3: Interpret t - coordinate

The t - coordinate (2) represents the time in seconds after launch when the rocket reaches its maximum height.

Step4: Interpret h - coordinate

The h - coordinate (64) represents the maximum height of the rocket in feet, which is 64 feet.

Answer:

The x - coordinate (or t - coordinate) of the vertex is \( 2 \) and represents the time (in seconds) when the rocket reaches its maximum height.
The y - coordinate (or h - coordinate) of the vertex is \( 64 \) and represents the maximum height (in feet) of the rocket.