If A Central Angle Of $\frac{5 \pi}{3}$ Is Created With Two Radii That Are 30 Inches Long, Then How Long Is The Arc They Will Cut In Radians?A. $\frac{\pi}{2}$ B. \$50 \pi$[/tex\] C. $\frac{5 \pi}{18}$

Introduction

In geometry, the relationship between central angles and arc lengths is a fundamental concept that helps us understand the properties of circles and their applications in various mathematical and real-world problems. A central angle is an angle formed by two radii that intersect at the center of a circle. In this article, we will explore how to find the length of an arc cut by two radii that form a central angle of $\frac{5 \pi}{3}$ radians, given that the radii are 30 inches long.

The Relationship Between Central Angles and Arc Lengths

The relationship between central angles and arc lengths is based on the concept of proportionality. The length of an arc is directly proportional to the measure of the central angle that subtends it. Mathematically, this can be expressed as:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the measure of the central angle in degrees, $r$ is the radius of the circle, and $2\pi r$ is the circumference of the circle.

Converting the Central Angle to Degrees

To apply the formula above, we need to convert the central angle from radians to degrees. We can do this using the following conversion factor:

$$1 \text{ radian} = \frac{180}{\pi} \text{ degrees}$$

Using this conversion factor, we can convert the central angle of $\frac{5 \pi}{3}$ radians to degrees as follows:

$$\frac{5 \pi}{3} \text{ radians} \times \frac{180}{\pi} \text{ degrees/radian} = 300 \text{ degrees}$$

Applying the Formula

Now that we have the central angle in degrees, we can apply the formula above to find the length of the arc:

$$\text{Arc Length} = \frac{300}{360} \times 2\pi \times 30$$

Simplifying the expression, we get:

$$\text{Arc Length} = \frac{5}{6} \times 2\pi \times 30$$

$$\text{Arc Length} = \frac{5}{6} \times 60\pi$$

$$\text{Arc Length} = 50\pi$$

Conclusion

In conclusion, the length of the arc cut by two radii that form a central angle of $\frac{5 \pi}{3}$ radians, given that the radii are 30 inches long, is $50\pi$ inches.

Answer

The correct answer is:

• A. $\frac{\pi}{2}$ (Incorrect)
• B. $50 \pi$ (Correct)
• C. $\frac{5 \pi}{18}$ (Incorrect)

Final Thoughts

Q: What is a central angle?

A: A central angle is an angle formed by two radii that intersect at the center of a circle.

Q: How is the length of an arc related to the measure of the central angle?

A: The length of an arc is directly proportional to the measure of the central angle that subtends it.

Q: What is the formula for finding the length of an arc?

A: The formula for finding the length of an arc is:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the measure of the central angle in degrees, $r$ is the radius of the circle, and $2\pi r$ is the circumference of the circle.

Q: How do I convert a central angle from radians to degrees?

A: To convert a central angle from radians to degrees, you can use the following conversion factor:

$$1 \text{ radian} = \frac{180}{\pi} \text{ degrees}$$

Q: What is the relationship between the central angle and the arc length in terms of radians?

A: The relationship between the central angle and the arc length in terms of radians is:

$$\text{Arc Length} = \theta \times r$$

where $\theta$ is the measure of the central angle in radians, and $r$ is the radius of the circle.

Q: How do I find the length of an arc if I know the central angle in radians and the radius of the circle?

A: To find the length of an arc if you know the central angle in radians and the radius of the circle, you can use the formula:

$$\text{Arc Length} = \theta \times r$$

Q: What is the significance of the central angle in determining the length of an arc?

A: The central angle is a crucial factor in determining the length of an arc. The larger the central angle, the longer the arc.

Q: Can the central angle be greater than 360 degrees?

A: Yes, the central angle can be greater than 360 degrees. However, in such cases, the arc length will be greater than the circumference of the circle.

Q: What is the relationship between the central angle and the arc length in terms of the circumference of the circle?

A: The relationship between the central angle and the arc length in terms of the circumference of the circle is:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the measure of the central angle in degrees, and $2\pi r$ is the circumference of the circle.

Q: How do I find the length of an arc if I know the central angle in degrees and the radius of the circle?

A: To find the length of an arc if you know the central angle in degrees and the radius of the circle, you can use the formula:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

Q: What is the significance of the radius in determining the length of an arc?

A: The radius is a crucial factor in determining the length of an arc. The larger the radius, the longer the arc.

Q: Can the radius be negative?

A: No, the radius cannot be negative. The radius is a measure of the distance from the center of the circle to the circumference, and it must always be a positive value.

Q: What is the relationship between the central angle and the arc length in terms of the radius and the circumference of the circle?

A: The relationship between the central angle and the arc length in terms of the radius and the circumference of the circle is:

$$\text{Arc Length} = \frac{\theta}{360} \times 2\pi r$$

where $\theta$ is the measure of the central angle in degrees, $r$ is the radius of the circle, and $2\pi r$ is the circumference of the circle.