Question

Fawzia launches a toy rocket from a platform. The height of the rocket in feet is given by h, of, t, equals, minus, 16, t, squared, plus, 88, t, plus, 48h(t)=−16t
2
+88t+48 where tt represents the time in seconds after launch. Interpret the coordinates of the vertex in context.

Answer 1

The height of the toy rocket is given by the quadratic function:

\[
h(t) = -16t^2 + 88t + 48
\]

To find the vertex of this quadratic function, we first identify that the vertex form of a quadratic function \( h(t) = at^2 + bt + c \) can be found using the formula for the time \( t \) at which the vertex occurs:

\[
t = -\frac{b}{2a}
\]

Where:
- \( a = -16 \)
- \( b = 88 \)

Substituting the values in:

\[
t = -\frac{88}{2 \times -16} = -\frac{88}{-32} = \frac{88}{32} = 2.75 \text{ seconds}
\]

Next, we find the height \( h(t) \) at this time by substituting \( t = 2.75 \) back into the height function:

\[
h(2.75) = -16(2.75)^2 + 88(2.75) + 48
\]

Calculating \( (2.75)^2 \):

\[
(2.75)^2 = 7.5625
\]

Now substituting this back into the height function:

\[
h(2.75) = -16(7.5625) + 88(2.75) + 48
\]
\[
= -121 + 242 + 48
\]
\[
= 169 \text{ feet}
\]

Thus, the coordinates of the vertex are \((2.75, 169)\).

### Interpretation in Context:
- The \(x\)-coordinate of the vertex, \(t = 2.75\) seconds, represents the time after launch when the rocket reaches its maximum height.
- The \(y\)-coordinate of the vertex, \(h = 169\) feet, represents the maximum height attained by the rocket during its flight.

In summary, the toy rocket reaches its highest point at \(2.75\) seconds after launch, achieving a maximum height of \(169\) feet.