Question

Jamal 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, 72, t, plus, 88h(t)=−16t
2
+72t+88 where tt represents the time in seconds after launch. What is the rocket’s greatest height?

Answer 1

To find the rocket's greatest height, we need to determine the vertex of the parabolic function given by

\[ h(t) = -16t^2 + 72t + 88. \]

The vertex of a parabola in the form \( h(t) = at^2 + bt + c \) can be found using the formula for the time at the vertex:

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

In our case, \( a = -16 \) and \( b = 72 \).

Substituting these values into the formula gives:

\[
t = -\frac{72}{2 \times -16} = -\frac{72}{-32} = \frac{72}{32} = \frac{9}{4} = 2.25 \text{ seconds}.
\]

Now we need to substitute \( t = 2.25 \) back into the height function \( h(t) \) to find the greatest height:

\[
h(2.25) = -16(2.25)^2 + 72(2.25) + 88.
\]

Calculating \( (2.25)^2 \):

\[
(2.25)^2 = 5.0625,
\]

Then, substituting that into the equation:

\[
h(2.25) = -16(5.0625) + 72(2.25) + 88.
\]

Calculating \( -16(5.0625) \):

\[
-16 \times 5.0625 = -81.
\]

Now calculating \( 72 \times 2.25 \):

\[
72 \times 2.25 = 162.
\]

Now we can assemble it all together:

\[
h(2.25) = -81 + 162 + 88 = 81 + 88 = 169.
\]

Thus, the greatest height of the rocket is

\[
\boxed{169} \text{ feet.}
\]