A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot.
y, equals, minus, 16, x, squared, plus, 170, x, plus, 85
y=−16x 
2
 +170x+85

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! 🚀

A rocket is launched from a tower. The height of the rocket, y in feet, is related to the time after launch, x in seconds, by the given equation. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot. y = −16x^2 + 170x + 85

Learn how to find the maximum height reached by a rocket using the quadratic equation y = -16x^2 + 170x + 85. This step-by-step solution uses the vertex formula to calculate the rocket's peak height of 535.4 feet, rounded to the nearest tenth. Suitable for students in Grades 8-10.

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