The given equation for the height of the rocket is:

$y = -16x^2 + 170x + 85$

This is a quadratic equation in the standard form $y = ax^2 + bx + c$, where:

• $a = -16$
• $b = 170$
• $c = 85$

Step 1: Find the Time at Maximum Height

The maximum height of a quadratic function $y = ax^2 + bx + c$ occurs at the vertex, given by:

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

Substituting the values:

$x = \frac{-170}{2(-16)}$

$x = \frac{170}{32} = 5.3125 \text{ seconds}$

Step 2: Calculate the Maximum Height

Now, substitute $x = 5.3125$ into the height equation:

$y = -16(5.3125)^2 + 170(5.3125) + 85$

$y = -16(28.2344) + 170(5.3125) + 85$

$y = -451.75 + 902.125 + 85$

$y = 535.375$

Final Answer:

The maximum height reached by the rocket is 535.4 feet (rounded to the nearest tenth).

Would you like a step-by-step breakdown of any part?

Here are some related questions to explore:

1. How long does the rocket take to reach the ground?
2. What is the initial height of the rocket?
3. What is the total flight time of the rocket?
4. At what time does the rocket reach a height of 400 feet?
5. How does air resistance affect the rocket's motion?

Tip: The vertex formula is useful for finding maximum or minimum values in quadratic functions! 🚀