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, 254, x, plus, 90
y=−16x 
2
 +254x+90

The height of the rocket is modeled by the quadratic equation:

\[
y = -16x^2 + 254x + 90
\]

Here, \(y\) is the height in feet, and \(x\) is the time in seconds. To find the maximum height, we need to determine the vertex of the parabola represented by this equation. For a quadratic equation of the form:

\[
y = ax^2 + bx + c
\]

the time \(x\) at which the maximum height occurs can be found using the formula:

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

For our equation:
- \(a = -16\)
- \(b = 254\)

Let's calculate the time \(x\) at which the rocket reaches its maximum height.

\[
x = \frac{-254}{2(-16)} = \frac{-254}{-32} = 7.9375 \, \text{seconds}
\]

Now, substitute \(x = 7.9375\) back into the original equation to find the maximum height \(y\).

\[
y = -16(7.9375)^2 + 254(7.9375) + 90
\]

Let me calculate that.The rocket reaches its maximum height at approximately \(7.9\) seconds, and the maximum height is \(1098.1\) feet (to the nearest tenth).

Would you like more details or have any further questions?

Here are five related questions to explore:
1. How would the maximum height change if the initial velocity were increased?
2. What would happen to the maximum height if the acceleration due to gravity was lower?
3. How long does the rocket stay in the air before hitting the ground?
4. What is the rocket's velocity at the maximum height?
5. How would you graph the rocket's trajectory over time?

**Tip:** The vertex formula \(\frac{-b}{2a}\) is a useful tool for quickly finding maximums or minimums of quadratic equations in physics.

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: y = −16x^2 + 254x + 90. Using this equation, find the maximum height reached by the rocket, to the nearest tenth of a foot.

Discover how to find the maximum height of a rocket launched from a tower using the quadratic equation y = −16x^2 + 254x + 90. This problem highlights key concepts in algebra, including maxima and minima, parabolas, and the vertex formula, suitable for students in Grades 9-11.

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