Consider Circle Y With A Radius Of 3 M And Central Angle XYZ Measuring $70^{\circ}$.What Is The Approximate Length Of Minor Arc XZ? Round To The Nearest Tenth Of A Meter.A. 1.8 Meters B. 3.7 Meters C. 15.2 Meters D. 18.8 Meters

Introduction

In geometry, a circle is a set of points that are all equidistant from a central point called the center. The distance from the center to any point on the circle is called the radius. A central angle is an angle formed by two radii that intersect at the center of the circle. In this article, we will explore how to calculate the length of a minor arc in a circle given the radius and the central angle.

Understanding the Problem

We are given a circle with a radius of 3 m and a central angle XYZ measuring $70^{\circ}$. The problem asks us to find the approximate length of minor arc XZ, rounded to the nearest tenth of a meter.

The Formula for the Length of a Minor Arc

The length of a minor arc in a circle can be calculated using the formula:

$$L = \frac{\theta}{360} \times 2\pi r$$

where:

• $L$ is the length of the minor arc
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Applying the Formula

In this case, we are given the radius $r = 3$ m and the central angle $\theta = 70^{\circ}$. Plugging these values into the formula, we get:

$$L = \frac{70}{360} \times 2\pi \times 3$$

Simplifying the Expression

To simplify the expression, we can first calculate the value of $\frac{70}{360}$:

$$\frac{70}{360} = \frac{7}{36}$$

Next, we can multiply this value by $2\pi$ and $3$:

$$L = \frac{7}{36} \times 2\pi \times 3$$

$$L = \frac{7}{36} \times 6\pi$$

$$L = \frac{7}{6} \pi$$

Evaluating the Expression

To evaluate the expression, we can use the value of $\pi \approx 3.14159$:

$$L = \frac{7}{6} \times 3.14159$$

$$L \approx 3.671$$

Rounding to the Nearest Tenth

Finally, we need to round the value of $L$ to the nearest tenth of a meter:

$$L \approx 3.7$$

Conclusion

In this article, we have shown how to calculate the length of a minor arc in a circle given the radius and the central angle. We have applied the formula for the length of a minor arc and simplified the expression to obtain the final answer. The approximate length of minor arc XZ is 3.7 meters.

Answer

The correct answer is:

• B. 3.7 meters

Discussion

This problem requires a good understanding of the formula for the length of a minor arc and the ability to apply it to a given situation. It also requires the use of mathematical concepts such as central angles and radii. The problem is relevant to mathematics and geometry, and it can be used to assess a student's understanding of these concepts.

Related Topics

• Central angles and radii
• Circumference and arc length
• Geometry and trigonometry
• Mathematical formulas and calculations

References

• [1] "Geometry" by Michael Artin
• [2] "Trigonometry" by Charles P. McKeague
• [3] "Mathematics for Engineers" by John R. Hubbard

Additional Resources

• Online calculators and tools for calculating arc length
• Interactive geometry software and apps
• Online resources and tutorials for learning geometry and trigonometry
  Q&A: Calculating the Length of a Minor Arc in a Circle =====================================================

Introduction

In our previous article, we explored how to calculate the length of a minor arc in a circle given the radius and the central angle. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for the length of a minor arc?

A: The formula for the length of a minor arc in a circle is:

$$L = \frac{\theta}{360} \times 2\pi r$$

where:

• $L$ is the length of the minor arc
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Q: What is the central angle?

A: The central angle is an angle formed by two radii that intersect at the center of the circle. It is measured in degrees and is used to calculate the length of a minor arc.

Q: What is the radius of the circle?

A: The radius of the circle is the distance from the center of the circle to any point on the circle. It is a key component in calculating the length of a minor arc.

Q: How do I calculate the length of a minor arc if I only know the circumference of the circle?

A: If you know the circumference of the circle, you can use the formula:

$$L = \frac{\theta}{360} \times C$$

where:

• $L$ is the length of the minor arc
• $\theta$ is the central angle in degrees
• $C$ is the circumference of the circle

Q: What is the difference between a minor arc and a major arc?

A: A minor arc is a portion of the circle that is less than 180 degrees. A major arc is a portion of the circle that is greater than 180 degrees.

Q: Can I use the formula for the length of a minor arc to calculate the length of a major arc?

A: Yes, you can use the formula for the length of a minor arc to calculate the length of a major arc. However, you will need to use the formula:

$$L = \frac{360 - \theta}{360} \times 2\pi r$$

where:

• $L$ is the length of the major arc
• $\theta$ is the central angle in degrees
• $r$ is the radius of the circle

Q: How do I round my answer to the nearest tenth of a meter?

A: To round your answer to the nearest tenth of a meter, you can use the following steps:

1. Calculate the length of the minor arc using the formula.
2. Round the answer to the nearest tenth of a meter.

Conclusion

In this article, we have answered some frequently asked questions related to calculating the length of a minor arc in a circle. We have provided formulas and explanations to help you understand the concept and apply it to real-world problems.

Additional Resources

• Online calculators and tools for calculating arc length
• Interactive geometry software and apps
• Online resources and tutorials for learning geometry and trigonometry

References

• [1] "Geometry" by Michael Artin
• [2] "Trigonometry" by Charles P. McKeague
• [3] "Mathematics for Engineers" by John R. Hubbard

Q&A Discussion

• What are some common mistakes to avoid when calculating the length of a minor arc?
• How do you handle cases where the central angle is greater than 180 degrees?
• Can you provide examples of real-world applications of calculating the length of a minor arc?

Related Topics

• Central angles and radii
• Circumference and arc length
• Geometry and trigonometry
• Mathematical formulas and calculations

Answer Key

• Q1: The formula for the length of a minor arc is $L = \frac{\theta}{360} \times 2\pi r$.
• Q2: The central angle is an angle formed by two radii that intersect at the center of the circle.
• Q3: The radius of the circle is the distance from the center of the circle to any point on the circle.
• Q4: You can use the formula $L = \frac{\theta}{360} \times C$ to calculate the length of a minor arc if you only know the circumference of the circle.
• Q5: A minor arc is a portion of the circle that is less than 180 degrees.
• Q6: Yes, you can use the formula for the length of a minor arc to calculate the length of a major arc.
• Q7: To round your answer to the nearest tenth of a meter, you can use the following steps: 1. Calculate the length of the minor arc using the formula. 2. Round the answer to the nearest tenth of a meter.