To find the time it takes for the rocket to reach its maximum height and the maximum height itself, we can analyze the given quadratic function:

$h(t) = -5t^2 + 90t$

Part 1: Time to Reach Maximum Height

The time at which the maximum height occurs in a quadratic function of the form $h(t) = at^2 + bt + c$ can be found using the formula:

$t = -\frac{b}{2a}$

Here, $a = -5$ and $b = 90$.

Plugging in the values:

$t = -\frac{90}{2 \times -5} = \frac{90}{10} = 9 \text{ seconds}$

Part 2: Maximum Height

To find the maximum height, substitute $t = 9$ back into the height function:

$h(9) = -5(9)^2 + 90(9)$

Calculating this step-by-step:

1. $9^2 = 81$
2. $-5 \times 81 = -405$
3. $90 \times 9 = 810$

Now, combine these results:

$h(9) = -405 + 810 = 405 \text{ meters}$

Summary of Results

• The rocket reaches the maximum height at 9 seconds.
• The maximum height that the rocket will reach is 405 meters.

If you have any further questions or need more details, feel free to ask! Here are some related questions you might consider:

1. How would the height function change if the initial velocity were different?
2. What happens to the time to reach maximum height if the acceleration due to gravity is altered?
3. How can we determine the total time the rocket is in the air?
4. What factors affect the rocket's trajectory and maximum height?
5. How can this model be applied to real-life rocket launches?

Tip: When analyzing quadratic functions, remember that the vertex represents the maximum or minimum point, which can be found using the vertex formula.