Question

A rocket is launched in the air. Its height in feet is given by h, equals, minus, 16, t, squared, plus, 136, th=−16t
2
+136t where tt represents the time in seconds after launch. Interpret the coordinates of the vertex in context.

Answer 1

To interpret the coordinates of the vertex of the quadratic function representing the rocket's height, we first need to rewrite the equation you provided:

\[ h(t) = -16t^2 + 136t \]

This is a quadratic equation in the standard form \( h(t) = at^2 + bt + c \), where:
- \( a = -16 \)
- \( b = 136 \)
- \( c = 0 \) (there's no constant term)

To find the vertex of the parabola, we can use the formula for the \( t \)-coordinate of the vertex, given by:

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

Substituting in the values of \( a \) and \( b \):

\[ t = -\frac{136}{2(-16)} = -\frac{136}{-32} = 4.25 \]

Now, we substitute \( t = 4.25 \) back into the height equation to find the height at this time:

\[ h(4.25) = -16(4.25)^2 + 136(4.25) \]
\[ = -16(18.0625) + 578 \]
\[ = -288 + 578 \]
\[ = 290 \]

So, the coordinates of the vertex are \( (4.25, 290) \).

### Interpretation in Context:

- **\( t = 4.25 \)** seconds: This is the time at which the rocket reaches its maximum height.
- **\( h = 290 \)** feet: This is the maximum height that the rocket reaches during its flight.

### Conclusion:

The vertex of the parabola indicates that 4.25 seconds after launch, the rocket reaches a maximum height of 290 feet before it begins to descend.