How Long Is The Arc Intersected By A Central Angle Of $\frac{\pi}{3}$ Radians In A Circle With A Radius Of 6 Ft? Round Your Answer To The Nearest Tenth. Use 3.14 For $\pi$.A. 1.0 Ft B. 5.7 Ft C. 6.3 Ft D. 7.0 Ft

by ADMIN 221 views

Introduction

In this article, we will delve into the world of mathematics, specifically geometry, to solve a problem involving a circle and a central angle. We will explore the relationship between the central angle, the radius of the circle, and the length of the arc intersected by the central angle. Our goal is to find the length of the arc intersected by a central angle of $\frac{\pi}{3}$ radians in a circle with a radius of 6 ft.

The Relationship Between Central Angle and Arc Length

The length of an arc in a circle is directly proportional to the radius of the circle and the measure of the central angle in radians. This relationship can be expressed using the formula:

Arc Length=Radius×Central Angle\text{Arc Length} = \text{Radius} \times \text{Central Angle}

In this case, we are given the radius of the circle as 6 ft and the central angle as $\frac{\pi}{3}$ radians. We will use the value of $\pi$ as 3.14 to calculate the length of the arc.

Calculating the Arc Length

Using the formula above, we can calculate the length of the arc intersected by the central angle:

Arc Length=6×π3\text{Arc Length} = 6 \times \frac{\pi}{3}

Substituting the value of $\pi$ as 3.14, we get:

Arc Length=6×3.143\text{Arc Length} = 6 \times \frac{3.14}{3}

Simplifying the expression, we get:

Arc Length=6×1.047\text{Arc Length} = 6 \times 1.047

Multiplying the values, we get:

Arc Length=6.282\text{Arc Length} = 6.282

Rounding the answer to the nearest tenth, we get:

Arc Length=6.3\text{Arc Length} = 6.3

Conclusion

In this article, we explored the relationship between the central angle, the radius of the circle, and the length of the arc intersected by the central angle. We used the formula $\text{Arc Length} = \text{Radius} \times \text{Central Angle}$ to calculate the length of the arc intersected by a central angle of $\frac{\pi}{3}$ radians in a circle with a radius of 6 ft. Our calculation resulted in an arc length of 6.3 ft, which is the correct answer.

Answer Key

The correct answer is C. 6.3 ft.

Additional Resources

For more information on geometry and trigonometry, please refer to the following resources:

Introduction

In our previous article, we explored the relationship between the central angle, the radius of the circle, and the length of the arc intersected by the central angle. We used the formula $\text{Arc Length} = \text{Radius} \times \text{Central Angle}$ to calculate the length of the arc intersected by a central angle of $\frac{\pi}{3}$ radians in a circle with a radius of 6 ft. In this article, we will answer some frequently asked questions related to arc length and central angles.

Q&A

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

A: The length of an arc in a circle is directly proportional to the radius of the circle and the measure of the central angle in radians. This relationship can be expressed using the formula:

Arc Length=Radius×Central Angle\text{Arc Length} = \text{Radius} \times \text{Central Angle}

Q: How do I calculate the arc length if I know the radius and the central angle?

A: To calculate the arc length, you can use the formula:

Arc Length=Radius×Central Angle\text{Arc Length} = \text{Radius} \times \text{Central Angle}

Make sure to use the correct units for the radius and the central angle. If the central angle is given in degrees, you will need to convert it to radians before using the formula.

Q: What is the difference between a central angle and an inscribed angle?

A: A central angle is an angle whose vertex is at the center of a circle, while an inscribed angle is an angle whose vertex is on the circumference of a circle. The measure of a central angle is equal to the measure of the arc it intersects, while the measure of an inscribed angle is half the measure of the arc it intersects.

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

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

Radian Measure=Degree Measure180×π\text{Radian Measure} = \frac{\text{Degree Measure}}{180} \times \pi

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

A: The arc length is a fraction of the circumference of a circle. The fraction is equal to the ratio of the central angle to 360 degrees (or $2\pi$ radians).

Q: How do I calculate the circumference of a circle?

A: To calculate the circumference of a circle, you can use the formula:

Circumference=2π×Radius\text{Circumference} = 2\pi \times \text{Radius}

Q: What is the relationship between the central angle and the area of a sector?

A: The area of a sector is directly proportional to the central angle and the radius of the circle. The relationship can be expressed using the formula:

Area of Sector=Central Angle360×π×Radius2\text{Area of Sector} = \frac{\text{Central Angle}}{360} \times \pi \times \text{Radius}^2

Q: How do I calculate the area of a sector?

A: To calculate the area of a sector, you can use the formula:

Area of Sector=Central Angle360×π×Radius2\text{Area of Sector} = \frac{\text{Central Angle}}{360} \times \pi \times \text{Radius}^2

Make sure to use the correct units for the radius and the central angle.

Conclusion

In this article, we answered some frequently asked questions related to arc length and central angles. We hope that this article has provided you with a better understanding of the relationship between the central angle, the radius of the circle, and the length of the arc intersected by the central angle. If you have any further questions, please don't hesitate to ask.

Additional Resources

For more information on geometry and trigonometry, please refer to the following resources: