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

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Introduction

Understanding the Problem The height of a rocket, y in feet, is related to the time after launch, x in seconds, by a given equation. In this article, we will use this equation to find the maximum height reached by the rocket, to the nearest tenth of a foot. This problem involves using mathematical concepts to analyze the motion of the rocket and determine its maximum height.

The Equation of Motion

The equation of motion for the rocket is given by:

y = -16x^2 + 256x

where y is the height of the rocket in feet and x is the time after launch in seconds.

Understanding the Equation

  • The equation is a quadratic equation, which represents a parabola when graphed.
  • The coefficient of x^2 is -16, which indicates that the parabola opens downward, meaning that the height of the rocket decreases as time increases.
  • The coefficient of x is 256, which indicates that the parabola has a maximum point, which represents the maximum height reached by the rocket.

Finding the Maximum Height

To find the maximum height reached by the rocket, we need to find the x-coordinate of the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

where a is the coefficient of x^2 and b is the coefficient of x.

Calculating the x-coordinate of the Vertex

In this case, a = -16 and b = 256. Plugging these values into the formula, we get:

x = -256 / (2 * -16) x = -256 / (-32) x = 8

Finding the Maximum Height

Now that we have the x-coordinate of the vertex, we can find the maximum height reached by the rocket by plugging this value into the equation of motion:

y = -16x^2 + 256x y = -16(8)^2 + 256(8) y = -16(64) + 2048 y = -1024 + 2048 y = 1024

Conclusion

The maximum height reached by the rocket is 1024 feet, to the nearest tenth of a foot. This problem demonstrates the use of mathematical concepts to analyze the motion of a rocket and determine its maximum height.

Discussion

  • The equation of motion for the rocket is a quadratic equation, which represents a parabola when graphed.
  • The parabola opens downward, meaning that the height of the rocket decreases as time increases.
  • The x-coordinate of the vertex of the parabola represents the time at which the rocket reaches its maximum height.
  • The maximum height reached by the rocket can be found by plugging the x-coordinate of the vertex into the equation of motion.

Additional Information

  • The equation of motion for the rocket can be used to determine the height of the rocket at any given time.
  • The parabola can be graphed to visualize the motion of the rocket.
  • The x-coordinate of the vertex can be used to determine the time at which the rocket reaches its maximum height.

References

  • [1] "Equations of Motion" by Math Open Reference
  • [2] "Quadratic Equations" by Math Is Fun
  • [3] "Graphing Quadratic Equations" by Purplemath

Related Topics

  • [1] "Motion under Gravity" by Khan Academy
  • [2] "Quadratic Equations and Functions" by Mathway
  • [3] "Graphing Quadratic Equations" by IXL

Introduction

In our previous article, we explored the equation of motion for a rocket launched from a tower and found the maximum height reached by the rocket. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q1: What is the equation of motion for the rocket?

A1: The equation of motion for the rocket is given by:

y = -16x^2 + 256x

where y is the height of the rocket in feet and x is the time after launch in seconds.

Q2: What does the equation represent?

A2: The equation represents a parabola when graphed, which means that the height of the rocket decreases as time increases.

Q3: How do I find the maximum height reached by the rocket?

A3: To find the maximum height reached by the rocket, you need to find the x-coordinate of the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

x = -b / 2a

where a is the coefficient of x^2 and b is the coefficient of x.

Q4: What is the x-coordinate of the vertex of the parabola?

A4: In this case, a = -16 and b = 256. Plugging these values into the formula, we get:

x = -256 / (2 * -16) x = -256 / (-32) x = 8

Q5: What is the maximum height reached by the rocket?

A5: Now that we have the x-coordinate of the vertex, we can find the maximum height reached by the rocket by plugging this value into the equation of motion:

y = -16x^2 + 256x y = -16(8)^2 + 256(8) y = -16(64) + 2048 y = -1024 + 2048 y = 1024

Q6: How do I graph the equation of motion?

A6: To graph the equation of motion, you can use a graphing calculator or a computer program. The graph will be a parabola that opens downward, meaning that the height of the rocket decreases as time increases.

Q7: What is the significance of the x-coordinate of the vertex?

A7: The x-coordinate of the vertex represents the time at which the rocket reaches its maximum height.

Q8: Can I use the equation of motion to determine the height of the rocket at any given time?

A8: Yes, you can use the equation of motion to determine the height of the rocket at any given time by plugging the value of x into the equation.

Conclusion

In this article, we have answered some frequently asked questions related to the equation of motion for a rocket launched from a tower. We have also provided additional information and resources for further learning.

Discussion

  • The equation of motion for the rocket is a quadratic equation, which represents a parabola when graphed.
  • The parabola opens downward, meaning that the height of the rocket decreases as time increases.
  • The x-coordinate of the vertex of the parabola represents the time at which the rocket reaches its maximum height.
  • The maximum height reached by the rocket can be found by plugging the x-coordinate of the vertex into the equation of motion.

Additional Information

  • The equation of motion for the rocket can be used to determine the height of the rocket at any given time.
  • The parabola can be graphed to visualize the motion of the rocket.
  • The x-coordinate of the vertex can be used to determine the time at which the rocket reaches its maximum height.

References

  • [1] "Equations of Motion" by Math Open Reference
  • [2] "Quadratic Equations" by Math Is Fun
  • [3] "Graphing Quadratic Equations" by Purplemath

Related Topics

  • [1] "Motion under Gravity" by Khan Academy
  • [2] "Quadratic Equations and Functions" by Mathway
  • [3] "Graphing Quadratic Equations" by IXL