Fawzia launches a toy rocket from a platform. The height of the rocket in feet is given by \( h(t)=-16 t^{2}+88 t+48 \) where \( t \) represents the time in seconds after launch. Interpret the coordinates of the vertex in context. Answer The x-coordinate (or t-coordinate) of the vertex is \( \square \) and represents The y-coordinate (or h-coordinate) of the vertex is \( \square \) and represents Th

To interpret the coordinates of the vertex for the rocket's height function \( h(t) = -16t^{2} + 88t + 48 \), we'll first determine the vertex's coordinates and then explain their significance in the context of the rocket launch. ### Finding the Vertex The vertex of a quadratic function \( h(t) = at^{2} + bt + c \) can be found using the formulas: \[ t_{\text{vertex}} = -\frac{b}{2a} \] \[ h_{\text{vertex}} = h(t_{\text{vertex}}) \] Given: - \( a = -16 \) - \( b = 88 \) - \( c = 48 \) **Calculate the \( t \)-coordinate of the vertex:** \[ t_{\text{vertex}} = -\frac{88}{2(-16)} = -\frac{88}{-32} = 2.75 \text{ seconds} \] **Calculate the \( h \)-coordinate of the vertex:** \[ h(2.75) = -16(2.75)^{2} + 88(2.75) + 48 \] \[ = -16(7.5625) + 242 + 48 \] \[ = -121 + 242 + 48 \] \[ = 169 \text{ feet} \] ### Interpretation of the Vertex Coordinates - **The \( t \)-coordinate of the vertex is \( \mathbf{2.75} \) seconds and represents** the time after launch when the rocket reaches its maximum height. - **The \( h \)-coordinate of the vertex is \( \mathbf{169} \) feet and represents** the maximum height attained by the rocket during its flight. ### Summary \[ \begin{align*} \text{The } t\text{-coordinate of the vertex is } & 2.75 \text{ seconds and represents the time at which the rocket reaches its maximum height.} \\ \text{The } h\text{-coordinate of the vertex is } & 169 \text{ feet and represents the maximum height achieved by the rocket.} \end{align*} \]