The Expression $\pi(r-3)^2$ Represents The Area Covered By The Hour Hand On A Clock. Write A Polynomial In Standard Form That Represents This Area For One Rotation.

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Introduction

The hour hand on a clock is a crucial component that helps us tell time. However, have you ever wondered how much area it covers during one rotation? The expression $\pi(r-3)^2$ represents the area covered by the hour hand on a clock. In this article, we will delve into the mathematical analysis of this expression and derive a polynomial in standard form that represents this area for one rotation.

Understanding the Expression

The expression $\pi(r-3)^2$ represents the area covered by the hour hand on a clock. Here, $r$ is the radius of the circle that the hour hand traces during one rotation. The expression $\pi(r-3)^2$ can be expanded as follows:

π(r−3)2=π(r2−6r+9)\pi(r-3)^2 = \pi(r^2 - 6r + 9)

This expression represents the area covered by the hour hand on a clock. However, we need to derive a polynomial in standard form that represents this area for one rotation.

Deriving the Polynomial

To derive the polynomial, we need to expand the expression $\pi(r^2 - 6r + 9)$ further. We can do this by multiplying the terms inside the parentheses:

π(r2−6r+9)=πr2−6πr+9π\pi(r^2 - 6r + 9) = \pi r^2 - 6\pi r + 9\pi

This expression represents the area covered by the hour hand on a clock in terms of the radius $r$. However, we need to derive a polynomial in standard form that represents this area for one rotation.

Standard Form Polynomial

A polynomial in standard form is a polynomial where the terms are arranged in descending order of the exponent of the variable. In this case, the variable is $r$. To derive the polynomial in standard form, we need to rearrange the terms in the expression $\pi r^2 - 6\pi r + 9\pi$:

πr2−6πr+9π=π(r2−6r+9)\pi r^2 - 6\pi r + 9\pi = \pi(r^2 - 6r + 9)

However, we can rewrite the expression as:

πr2−6πr+9π=π(r2−6r)+9π\pi r^2 - 6\pi r + 9\pi = \pi(r^2 - 6r) + 9\pi

Now, we can factor out the common term $r$ from the first two terms:

πr2−6πr+9π=πr(r−6)+9π\pi r^2 - 6\pi r + 9\pi = \pi r(r - 6) + 9\pi

This expression represents the area covered by the hour hand on a clock in terms of the radius $r$ in standard form.

Conclusion

In this article, we have analyzed the expression $\pi(r-3)^2$ that represents the area covered by the hour hand on a clock. We have derived a polynomial in standard form that represents this area for one rotation. The polynomial is $\pi r(r - 6) + 9\pi$, where $r$ is the radius of the circle that the hour hand traces during one rotation. This polynomial provides a mathematical representation of the area covered by the hour hand on a clock in terms of the radius $r$.

Applications

The expression $\pi(r-3)^2$ and the polynomial $\pi r(r - 6) + 9\pi$ have several applications in mathematics and physics. For example, they can be used to calculate the area covered by the hour hand on a clock for different values of the radius $r$. They can also be used to derive formulas for the area covered by other objects that trace a circular path.

Future Research Directions

There are several future research directions that can be explored based on the expression $\pi(r-3)^2$ and the polynomial $\pi r(r - 6) + 9\pi$. For example, researchers can investigate the properties of the polynomial and its applications in different fields. They can also explore the relationship between the radius $r$ and the area covered by the hour hand on a clock.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak
  • [3] "Geometry" by I.M. Gelfand

Note: The references provided are for illustrative purposes only and are not actual references used in this article.

Introduction

In our previous article, we analyzed the expression $\pi(r-3)^2$ that represents the area covered by the hour hand on a clock. We also derived a polynomial in standard form that represents this area for one rotation. In this article, we will answer some frequently asked questions related to the expression and the polynomial.

Q&A

Q: What is the significance of the expression $\pi(r-3)^2$?

A: The expression $\pi(r-3)^2$ represents the area covered by the hour hand on a clock. It is a mathematical representation of the area covered by the hour hand for a given radius $r$.

Q: How is the expression $\pi(r-3)^2$ derived?

A: The expression $\pi(r-3)^2$ is derived by considering the area covered by the hour hand on a clock. The hour hand traces a circular path with a radius $r$, and the area covered by the hour hand is given by the formula $\pi(r-3)^2$.

Q: What is the polynomial in standard form that represents the area covered by the hour hand on a clock?

A: The polynomial in standard form that represents the area covered by the hour hand on a clock is $\pi r(r - 6) + 9\pi$.

Q: What is the significance of the term $9\pi$ in the polynomial?

A: The term $9\pi$ represents the constant area covered by the hour hand on a clock. It is a fixed value that does not depend on the radius $r$.

Q: How can the polynomial be used in real-world applications?

A: The polynomial can be used to calculate the area covered by the hour hand on a clock for different values of the radius $r$. It can also be used to derive formulas for the area covered by other objects that trace a circular path.

Q: What are some potential applications of the expression and the polynomial?

A: Some potential applications of the expression and the polynomial include:

  • Calculating the area covered by the hour hand on a clock for different values of the radius $r$
  • Deriving formulas for the area covered by other objects that trace a circular path
  • Investigating the properties of the polynomial and its applications in different fields
  • Exploring the relationship between the radius $r$ and the area covered by the hour hand on a clock

Q: What are some potential future research directions related to the expression and the polynomial?

A: Some potential future research directions related to the expression and the polynomial include:

  • Investigating the properties of the polynomial and its applications in different fields
  • Exploring the relationship between the radius $r$ and the area covered by the hour hand on a clock
  • Deriving formulas for the area covered by other objects that trace a circular path
  • Investigating the applications of the expression and the polynomial in real-world scenarios

Conclusion

In this article, we have answered some frequently asked questions related to the expression $\pi(r-3)^2$ and the polynomial $\pi r(r - 6) + 9\pi$. We hope that this article has provided a better understanding of the significance and applications of the expression and the polynomial.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak
  • [3] "Geometry" by I.M. Gelfand

Note: The references provided are for illustrative purposes only and are not actual references used in this article.