Question

A rocket is launched in the air. Its height in feet is given by h(t) = -16t^2 + 104t 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 _ and represents _____ ______

The y-coordinate (or h coordinate) of the Vertex is _ and represents _____ ______

Answer 1

To find the coordinates of the vertex of the parabola represented by the function \( h(t) = -16t^2 + 104t \), we first need to determine the vertex's \( t \)-coordinate, which is given by the formula:

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

In the function \( h(t) = -16t^2 + 104t \), we have \( a = -16 \) and \( b = 104 \).

Substituting these values into the formula gives:

\[
t = -\frac{104}{2 \cdot -16} = -\frac{104}{-32} = \frac{104}{32} = \frac{13}{4} = 3.25 \text{ seconds}
\]

### x-coordinate (or t coordinate) of the Vertex:
\[
t = 3.25
\]

### Interpretation:
This \( t \)-coordinate represents the time (in seconds) at which the rocket reaches its maximum height.

Next, we need to find the \( h \)-coordinate (or y-coordinate) of the vertex by substituting \( t = 3.25 \) back into the height function \( h(t) \):

\[
h(3.25) = -16(3.25)^2 + 104(3.25)
\]

Calculating \( (3.25)^2 \):

\[
(3.25)^2 = 10.5625
\]

Now substituting this in:

\[
h(3.25) = -16(10.5625) + 104(3.25)
\]
\[
= -169 + 338 = 169 \text{ feet}
\]

### y-coordinate (or h coordinate) of the Vertex:
\[
h = 169
\]

### Interpretation:
This \( h \)-coordinate represents the maximum height (in feet) that the rocket reaches after 3.25 seconds.

### Summary:
- The x-coordinate (or t coordinate) of the vertex is **3.25 seconds** and represents **the time at which the rocket reaches its maximum height**.
- The y-coordinate (or h coordinate) of the vertex is **169 feet** and represents **the maximum height reached by the rocket**.

Answer 2

Answer the questions please.