Question

(consider the vertex you found in the previous problem)
a rocket is launched in the air. its height in feet is given by ( h(t)=-16 t^{2}+96 t ) where ( t ) represents the time in seconds after launch. what does the ( t )-coordinate of the vertex mean?

• how long the rocket has been in the air at its peak
• how long the rocket has been in the air when it hits the ground
• the rockets height at its peak
• the rockets height when it hits the ground

Explanation:

The function \( h(t)=-16t^{2}+96t \) is a quadratic function in the form \( y = ax^{2}+bx + c \) (here \( y = h(t)\), \(x=t\), \(a=-16\), \(b = 96\), \(c = 0\)). For a quadratic function \(y=ax^{2}+bx + c\), the vertex of the parabola (since \(a=-16<0\), the parabola opens down - ward) gives the maximum value of the function. The \(t\) - coordinate of the vertex of the parabola \(y = ax^{2}+bx + c\) is given by \(t=-\frac{b}{2a}\). In the context of the height - time function \(h(t)\), \(t\) represents time. The vertex of the parabola \(h(t)\) gives the time at which the rocket reaches its maximum height (peak).

Answer:

how long the rocket has been in the air at it's peak