Math Problem Statement
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
Solution
The height of the rocket is modeled by the quadratic equation:
Here, is the height in feet, and 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:
the time at which the maximum height occurs can be found using the formula:
For our equation:
Let's calculate the time at which the rocket reaches its maximum height.
Now, substitute back into the original equation to find the maximum height .
Let me calculate that.The rocket reaches its maximum height at approximately seconds, and the maximum height is feet (to the nearest tenth).
Would you like more details or have any further questions?
Here are five related questions to explore:
- How would the maximum height change if the initial velocity were increased?
- What would happen to the maximum height if the acceleration due to gravity was lower?
- How long does the rocket stay in the air before hitting the ground?
- What is the rocket's velocity at the maximum height?
- How would you graph the rocket's trajectory over time?
Tip: The vertex formula is a useful tool for quickly finding maximums or minimums of quadratic equations in physics.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Maxima and Minima
Parabolas
Formulas
y = ax^2 + bx + c
Vertex formula: x = -b / 2a
Theorems
Quadratic formula
Properties of parabolas
Suitable Grade Level
Grades 9-11
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