In A Circle With A Radius Of 6 Ft, An Arc Is Intercepted By A Central Angle Of $\frac{3 \pi}{2}$ Radians.What Is The Length Of The Arc?A. \$2 \pi$ Ft$ B. $3 \pi$ Ft$ C. \$6 \pi$ Ft$ D.

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In a Circle with a Radius of 6 ft, an Arc is Intercepted by a Central Angle of $\frac{3 \pi}{2}$ Radians: What is the Length of the Arc?

Understanding the Problem

When dealing with circles and arcs, it's essential to understand the relationship between the central angle, the radius of the circle, and the length of the arc. In this problem, we're given a circle with a radius of 6 ft and a central angle of $\frac{3 \pi}{2}$ radians. Our goal is to find the length of the arc intercepted by this central angle.

Recalling the Formula for Arc Length

The formula for the length of an arc in a circle is given by:

L=θ×rL = \theta \times r

where:

  • L$ is the length of the arc

  • \theta$ is the central angle in radians

  • r$ is the radius of the circle

Applying the Formula

In this problem, we're given the central angle $\theta = \frac{3 \pi}{2}$ radians and the radius $r = 6$ ft. We can now plug these values into the formula to find the length of the arc:

L=3π2×6L = \frac{3 \pi}{2} \times 6

Simplifying the Expression

To simplify the expression, we can multiply the central angle by the radius:

L=3π2×6L = \frac{3 \pi}{2} \times 6

L=9πL = 9 \pi

Evaluating the Answer Choices

Now that we have the length of the arc, we can evaluate the answer choices:

  • A. $2 \pi$ ft
  • B. $3 \pi$ ft
  • C. $6 \pi$ ft
  • D. $9 \pi$ ft

Based on our calculation, the correct answer is:

  • C. $6 \pi$ ft is incorrect, the correct answer is D. $9 \pi$ ft

Conclusion

In this problem, we used the formula for arc length to find the length of the arc intercepted by a central angle of $\frac{3 \pi}{2}$ radians in a circle with a radius of 6 ft. By applying the formula and simplifying the expression, we found that the length of the arc is $9 \pi$ ft. This problem demonstrates the importance of understanding the relationship between the central angle, the radius of the circle, and the length of the arc in order to solve problems involving circles and arcs.

Additional Examples and Applications

This problem is a great example of how to apply the formula for arc length in a real-world scenario. Here are a few additional examples and applications:

  • Circular Motion: In physics, the formula for arc length is used to describe the motion of objects in circular paths. For example, the length of the arc traveled by a car on a circular track can be calculated using the formula.
  • Engineering: In engineering, the formula for arc length is used to design and optimize circular structures such as bridges, tunnels, and pipes.
  • Geometry: In geometry, the formula for arc length is used to calculate the length of arcs in various shapes and figures.

Common Mistakes to Avoid

When working with the formula for arc length, there are a few common mistakes to avoid:

  • Incorrect units: Make sure to use the correct units for the central angle and the radius. In this problem, the central angle is given in radians, and the radius is given in feet.
  • Incorrect calculation: Double-check your calculation to ensure that you have the correct answer.
  • Not considering the radius: Make sure to consider the radius of the circle when calculating the length of the arc.

Real-World Applications

The formula for arc length has numerous real-world applications in various fields such as:

  • Aerospace Engineering: In aerospace engineering, the formula for arc length is used to calculate the length of the arc traveled by a spacecraft in a circular orbit.
  • Civil Engineering: In civil engineering, the formula for arc length is used to design and optimize circular structures such as bridges, tunnels, and pipes.
  • Computer-Aided Design (CAD): In CAD, the formula for arc length is used to create and edit circular shapes and figures.

Conclusion

In conclusion, the formula for arc length is a fundamental concept in mathematics that has numerous real-world applications. By understanding the relationship between the central angle, the radius of the circle, and the length of the arc, we can solve problems involving circles and arcs. This problem demonstrates the importance of applying the formula for arc length in a real-world scenario and highlights the need to avoid common mistakes when working with this formula.
Q&A: Arc Length and Central Angles

Frequently Asked Questions

In this article, we'll answer some of the most frequently asked questions about arc length and central angles.

Q: What is the formula for arc length?

A: The formula for arc length is given by:

L=θ×rL = \theta \times r

where:

  • L$ is the length of the arc

  • \theta$ is the central angle in radians

  • r$ is the radius of the circle

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

A: The central angle and the length of the arc are directly proportional. As the central angle increases, the length of the arc also increases.

Q: What is the unit of measurement for the central angle?

A: The unit of measurement for the central angle is radians.

Q: What is the unit of measurement for the radius?

A: The unit of measurement for the radius is typically the same as the unit of measurement for the length of the arc.

Q: Can the central angle be greater than 2π radians?

A: Yes, the central angle can be greater than 2π radians. However, the length of the arc will be greater than the circumference of the circle.

Q: Can the central angle be less than 0 radians?

A: No, the central angle cannot be less than 0 radians.

Q: What is the relationship between the circumference of the circle and the length of the arc?

A: The circumference of the circle is equal to the length of the arc when the central angle is 2π radians.

Q: Can the length of the arc be greater than the circumference of the circle?

A: Yes, the length of the arc can be greater than the circumference of the circle if the central angle is greater than 2π radians.

Q: How do I calculate the length of the arc when the central angle is given in degrees?

A: To calculate the length of the arc when the central angle is given in degrees, you need to convert the central angle from degrees to radians using the formula:

θradians=θdegrees×π180\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}

Then, you can use the formula for arc length:

L=θradians×rL = \theta_{\text{radians}} \times r

Q: How do I calculate the length of the arc when the radius is given in a different unit of measurement?

A: To calculate the length of the arc when the radius is given in a different unit of measurement, you need to convert the radius to the same unit of measurement as the length of the arc.

Q: What are some real-world applications of arc length and central angles?

A: Some real-world applications of arc length and central angles include:

  • Aerospace Engineering: In aerospace engineering, arc length and central angles are used to calculate the length of the arc traveled by a spacecraft in a circular orbit.
  • Civil Engineering: In civil engineering, arc length and central angles are used to design and optimize circular structures such as bridges, tunnels, and pipes.
  • Computer-Aided Design (CAD): In CAD, arc length and central angles are used to create and edit circular shapes and figures.

Conclusion

In conclusion, arc length and central angles are fundamental concepts in mathematics that have numerous real-world applications. By understanding the relationship between the central angle, the radius of the circle, and the length of the arc, we can solve problems involving circles and arcs. This article has provided answers to some of the most frequently asked questions about arc length and central angles.