The Height Y Y Y , In Feet, Of A Football X X X Seconds After It Is Thrown Is Modeled By The Equation Y = − 16 X 2 + 24 X + 1 Y=-16x^2+24x+1 Y = − 16 X 2 + 24 X + 1 . What Information Does The Zero Of The Equation 0 = − 16 X 2 + 24 X + 1 0=-16x^2+24x+1 0 = − 16 X 2 + 24 X + 1 Represent?A. The Height Of The Ball

Introduction

When a football is thrown into the air, its height can be modeled using a quadratic equation. The equation $y=-16x^2+24x+1$ represents the height $y$, in feet, of the football $x$ seconds after it is thrown. In this article, we will explore the concept of the zero of a quadratic equation and what information it represents in the context of the football's height.

What is a Zero of a Quadratic Equation?

A zero of a quadratic equation is a value of $x$ that makes the equation equal to zero. In other words, it is a solution to the equation when the height of the football is zero. To find the zero of the equation $0=-16x^2+24x+1$, we need to solve for $x$.

Solving the Quadratic Equation

To solve the quadratic equation $0=-16x^2+24x+1$, we can use the quadratic formula:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

where $a=-16$, $b=24$, and $c=1$. Plugging these values into the formula, we get:

$x=\frac{-24\pm\sqrt{24^2-4(-16)(1)}}{2(-16)}$

Simplifying the expression under the square root, we get:

$x=\frac{-24\pm\sqrt{576+64}}{-32}$

$x=\frac{-24\pm\sqrt{640}}{-32}$

$x=\frac{-24\pm16\sqrt{10}}{-32}$

$x=\frac{3\pm2\sqrt{10}}{4}$

Interpreting the Zero of the Equation

The zero of the equation $0=-16x^2+24x+1$ represents the time at which the height of the football is zero. In other words, it is the time when the football hits the ground. The two solutions to the equation, $x=\frac{3+2\sqrt{10}}{4}$ and $x=\frac{3-2\sqrt{10}}{4}$, represent the two times when the football is at ground level.

Physical Significance of the Zero

The zero of the equation has significant physical implications. It represents the time when the football's height is zero, which is when the football hits the ground. This is an important aspect of the football's trajectory, as it determines the duration of the flight.

Conclusion

In conclusion, the zero of the equation $0=-16x^2+24x+1$ represents the time when the height of the football is zero, which is when the football hits the ground. The two solutions to the equation provide the two times when the football is at ground level. Understanding the zero of a quadratic equation is essential in modeling the height of a thrown football and predicting its trajectory.

Discussion

• What other information can be obtained from the quadratic equation $y=-16x^2+24x+1$?
• How does the zero of the equation relate to the football's velocity and acceleration?
• Can the quadratic equation be used to model the height of other objects, such as a thrown basketball or a falling object?

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "The Height of a Thrown Object" by Physics Classroom
• [3] "Quadratic Formula" by Wolfram MathWorld
  The Height of a Thrown Football: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the concept of the zero of a quadratic equation and its relation to the height of a thrown football. In this article, we will answer some frequently asked questions about the quadratic equation $y=-16x^2+24x+1$ and its application to the football's trajectory.

Q&A

Q: What is the significance of the quadratic equation $y=-16x^2+24x+1$?

A: The quadratic equation $y=-16x^2+24x+1$ represents the height $y$, in feet, of a football $x$ seconds after it is thrown. It is a mathematical model that describes the football's trajectory.

Q: What does the zero of the equation represent?

A: The zero of the equation represents the time when the height of the football is zero, which is when the football hits the ground.

Q: How do I find the zero of the equation?

A: To find the zero of the equation, you can use the quadratic formula:

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

where $a=-16$, $b=24$, and $c=1$. Plugging these values into the formula, you get:

$x=\frac{-24\pm\sqrt{24^2-4(-16)(1)}}{2(-16)}$

Simplifying the expression under the square root, you get:

$x=\frac{-24\pm\sqrt{576+64}}{-32}$

$x=\frac{-24\pm\sqrt{640}}{-32}$

$x=\frac{-24\pm16\sqrt{10}}{-32}$

$x=\frac{3\pm2\sqrt{10}}{4}$

Q: What is the physical significance of the zero of the equation?

A: The zero of the equation has significant physical implications. It represents the time when the football's height is zero, which is when the football hits the ground. This is an important aspect of the football's trajectory, as it determines the duration of the flight.

Q: Can the quadratic equation be used to model the height of other objects?

A: Yes, the quadratic equation can be used to model the height of other objects, such as a thrown basketball or a falling object. However, the equation would need to be adjusted to account for the object's specific characteristics, such as its mass and air resistance.

Q: How does the zero of the equation relate to the football's velocity and acceleration?

A: The zero of the equation is related to the football's velocity and acceleration through the concept of the derivative and the second derivative. The derivative of the equation represents the football's velocity, while the second derivative represents the football's acceleration.

Q: Can the quadratic equation be used to predict the football's trajectory?

A: Yes, the quadratic equation can be used to predict the football's trajectory. By plugging in different values of $x$, you can calculate the corresponding values of $y$, which represent the football's height at different times.

Conclusion

In conclusion, the quadratic equation $y=-16x^2+24x+1$ is a powerful tool for modeling the height of a thrown football. By understanding the zero of the equation and its physical significance, you can gain valuable insights into the football's trajectory and predict its behavior.

Discussion

• What other objects can be modeled using the quadratic equation?
• How does the zero of the equation relate to the football's spin and rotation?
• Can the quadratic equation be used to model the height of objects in different environments, such as on a planet or in a vacuum?

References

• [1] "Quadratic Equations" by Math Open Reference
• [2] "The Height of a Thrown Object" by Physics Classroom
• [3] "Quadratic Formula" by Wolfram MathWorld