The given problem is a quadratic function representing the height $h$ of a rocket as a function of time $t$, with the equation:

$h(t) = -4.9t^2 + 253t + 367$

This is a quadratic equation in the form $h(t) = at^2 + bt + c$, where:

• $a = -4.9$
• $b = 253$
• $c = 367$

To find the maximum height, we need to determine the vertex of the parabola, which gives the peak height. The formula to find the time $t$ at which the maximum height occurs (the vertex of the parabola) is:

$t_{\text{max}} = \frac{-b}{2a}$

Substituting the values of $a$ and $b$:

$t_{\text{max}} = \frac{-253}{2(-4.9)} = \frac{-253}{-9.8} = 25.82 \text{ seconds}$

Now, substitute this value of $t_{\text{max}}$ into the height function $h(t)$ to find the maximum height:

$h(25.82) = -4.9(25.82)^2 + 253(25.82) + 367$

$h(25.82) = -4.9(666.87) + 6531.46 + 367$

$h(25.82) = -3267.66 + 6531.46 + 367 = 3630.80 \text{ meters}$

Thus, the rocket reaches its peak height of 3630.80 meters above sea level.

Let me know if you'd like further details or if you have any questions!

Here are 5 related questions:

1. How do you find the vertex of a quadratic function in general?
2. What does the value of $a$ signify in the context of this rocket's motion?
3. How does the constant term $c$ in the quadratic equation affect the height function?
4. Can you calculate how long it takes for the rocket to hit the ground?
5. How would you modify the equation if air resistance were considered?

Tip: The vertex form of a quadratic function makes it easier to identify the peak or minimum value of the function directly!