If The Radius Of Circle $O$ Is 14 Meters And It Is Dilated By A Scale Factor Of 2.5, Then What Will Be The Circumference Of Circle $O^{\prime}$?A. 44 M B. 345 M C. 10 M D. 220 M
Introduction
In geometry, dilation is a transformation that changes the size of a figure. When a circle is dilated by a scale factor, its radius and circumference are affected. In this article, we will explore how to calculate the circumference of a circle after dilation and apply this concept to a specific problem.
What is Dilation?
Dilation is a transformation that changes the size of a figure by a scale factor. It is a type of similarity transformation that preserves the shape of the figure but changes its size. When a circle is dilated by a scale factor, its radius and circumference are affected.
Properties of Dilation
When a circle is dilated by a scale factor, the following properties hold:
- The center of the circle remains the same.
- The radius of the circle is multiplied by the scale factor.
- The circumference of the circle is multiplied by the scale factor.
Calculating Circumference After Dilation
To calculate the circumference of a circle after dilation, we can use the following formula:
Circumference = 2πr
where r is the radius of the circle.
When a circle is dilated by a scale factor, its radius is multiplied by the scale factor. Therefore, the new radius is:
r' = kr
where k is the scale factor.
The new circumference is:
C' = 2πr'
Substituting the expression for r', we get:
C' = 2π(kr)
C' = k(2πr)
Therefore, the circumference of the circle after dilation is:
C' = kC
where C is the original circumference.
Problem: Dilation of Circle O
The radius of circle O is 14 meters, and it is dilated by a scale factor of 2.5. What will be the circumference of circle O'?
Step 1: Calculate the Original Circumference
The original circumference of circle O is:
C = 2πr
where r is the radius of circle O.
Substituting the value of r, we get:
C = 2π(14)
C = 28π
Step 2: Calculate the New Circumference
The new circumference of circle O' is:
C' = kC
where k is the scale factor.
Substituting the value of k and C, we get:
C' = 2.5(28π)
C' = 70π
Step 3: Calculate the Numerical Value
To calculate the numerical value of the new circumference, we can use the value of π as approximately 3.14.
C' = 70(3.14)
C' = 220.8
Rounding to the nearest whole number, we get:
C' = 221 m
However, the closest answer choice is 220 m.
Conclusion
In this article, we explored how to calculate the circumference of a circle after dilation. We used the formula C' = kC to calculate the new circumference and applied this concept to a specific problem. The radius of circle O is 14 meters, and it is dilated by a scale factor of 2.5. The circumference of circle O' is approximately 220 m.
References
- [1] "Dilation" by Khan Academy
- [2] "Circumference of a Circle" by Math Open Reference
- [3] "Dilation of a Circle" by Purplemath
Discussion
Introduction
In our previous article, we explored how to calculate the circumference of a circle after dilation. We used the formula C' = kC to calculate the new circumference and applied this concept to a specific problem. In this article, we will answer some frequently asked questions about dilation and circumference calculation.
Q: What is dilation?
A: Dilation is a transformation that changes the size of a figure by a scale factor. It is a type of similarity transformation that preserves the shape of the figure but changes its size.
Q: What are the properties of dilation?
A: When a circle is dilated by a scale factor, the following properties hold:
- The center of the circle remains the same.
- The radius of the circle is multiplied by the scale factor.
- The circumference of the circle is multiplied by the scale factor.
Q: How do I calculate the circumference of a circle after dilation?
A: To calculate the circumference of a circle after dilation, you can use the formula C' = kC, where k is the scale factor and C is the original circumference.
Q: What is the formula for the circumference of a circle?
A: The formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.
Q: How do I calculate the new radius of a circle after dilation?
A: To calculate the new radius of a circle after dilation, you can multiply the original radius by the scale factor. The new radius is r' = kr.
Q: What is the relationship between the original and new circumferences of a circle after dilation?
A: The new circumference of a circle after dilation is equal to the original circumference multiplied by the scale factor. This can be expressed as C' = kC.
Q: Can you give an example of how to calculate the circumference of a circle after dilation?
A: Let's say we have a circle with a radius of 14 meters, and it is dilated by a scale factor of 2.5. To calculate the new circumference, we can use the formula C' = kC.
First, we calculate the original circumference:
C = 2πr
C = 2π(14)
C = 28π
Next, we multiply the original circumference by the scale factor:
C' = kC
C' = 2.5(28π)
C' = 70π
Finally, we calculate the numerical value of the new circumference using the value of π as approximately 3.14:
C' = 70(3.14)
C' = 220.8
Rounding to the nearest whole number, we get:
C' = 221 m
However, the closest answer choice is 220 m.
Q: What are some real-world applications of dilation and circumference calculation?
A: Dilation and circumference calculation have many real-world applications, including:
- Architecture: Dilation is used to design buildings and structures that are scaled up or down to fit specific needs.
- Engineering: Dilation is used to design machines and mechanisms that require precise scaling.
- Art: Dilation is used to create scaled-up or scaled-down versions of artwork.
- Science: Dilation is used to study the behavior of objects at different scales.
Conclusion
In this article, we answered some frequently asked questions about dilation and circumference calculation. We explored the properties of dilation, how to calculate the circumference of a circle after dilation, and some real-world applications of dilation and circumference calculation. We hope this article has been helpful in understanding these concepts.
References
- [1] "Dilation" by Khan Academy
- [2] "Circumference of a Circle" by Math Open Reference
- [3] "Dilation of a Circle" by Purplemath
Discussion
What are some other real-world applications of dilation and circumference calculation? How can you use these concepts to solve problems in your daily life? Share your thoughts and ideas in the comments below!