For A Circle Of Radius 3 Feet, Find The Arc Length $s$ Subtended By A Central Angle Of $57^{\circ}$.A. $s=\frac{19}{40} \pi$ Feet B. $s=\frac{19}{5} \pi$ Feet C. $s=\frac{19}{20} \pi$ Feet D.

Feb 27, 2025 by ADMIN 195 views

Introduction

In geometry, the arc length of a circle is the length of the curved line that makes up a part of the circle's circumference. Given a circle with a known radius and a central angle, we can calculate the arc length using the formula: $s = \frac{\theta}{360} \cdot 2\pi r$ , where $s$ is the arc length, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle. In this article, we will use this formula to find the arc length of a circle with a radius of 3 feet and a central angle of $57^{\circ}$ .

The Formula for Arc Length

The formula for arc length is given by: $s = \frac{\theta}{360} \cdot 2\pi r$ . This formula can be derived by considering the circle as being divided into 360 equal parts, each with a central angle of $1^{\circ}$ . The arc length of each part is then given by: $s = \frac{1}{360} \cdot 2\pi r$ . Multiplying this by the number of parts, we get the total arc length: $s = \frac{\theta}{360} \cdot 2\pi r$ .

Applying the Formula

Now that we have the formula, we can apply it to the given problem. We are given a circle with a radius of 3 feet and a central angle of $57^{\circ}$ . Plugging these values into the formula, we get: $s = \frac{57}{360} \cdot 2\pi \cdot 3$ .

Simplifying the Expression

To simplify the expression, we can start by multiplying the numbers together: $s = \frac{57}{360} \cdot 6\pi$ . Next, we can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 3: $s = \frac{19}{120} \cdot 6\pi$ . Finally, we can simplify the expression by canceling out the common factor of 6: $s = \frac{19}{20} \pi$ .

Conclusion

In conclusion, the arc length of a circle with a radius of 3 feet and a central angle of $57^{\circ}$ is given by: $s = \frac{19}{20} \pi$ feet. This can be calculated using the formula: $s = \frac{\theta}{360} \cdot 2\pi r$ , where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.

Comparison of Options

Now that we have calculated the arc length, we can compare our answer to the options given:

A. $s=\frac{19}{40} \pi$ feet
B. $s=\frac{19}{5} \pi$ feet
C. $s=\frac{19}{20} \pi$ feet
D. (no option)

Our answer, $s = \frac{19}{20} \pi$ feet, matches option C.

Discussion

The arc length of a circle is an important concept in geometry, and it has many practical applications in fields such as engineering and physics. In this article, we used the formula: $s = \frac{\theta}{360} \cdot 2\pi r$ to find the arc length of a circle with a radius of 3 feet and a central angle of $57^{\circ}$ . We simplified the expression and arrived at the answer: $s = \frac{19}{20} \pi$ feet. This answer matches option C, and it demonstrates the importance of using the correct formula and simplifying expressions in order to arrive at the correct answer.

Additional Examples

Here are a few additional examples of how to use the formula for arc length:

Find the arc length of a circle with a radius of 4 feet and a central angle of $30^{\circ}$ .
Find the arc length of a circle with a radius of 5 feet and a central angle of $45^{\circ}$ .
Find the arc length of a circle with a radius of 6 feet and a central angle of $60^{\circ}$ .

These examples demonstrate the versatility of the formula for arc length and how it can be used to solve a wide range of problems.

Conclusion

Q: What is the arc length of a circle?

A: The arc length of a circle is the length of the curved line that makes up a part of the circle's circumference.

Q: How do I calculate the arc length of a circle?

A: To calculate the arc length of a circle, you can use the formula: $s = \frac{\theta}{360} \cdot 2\pi r$ , where $s$ is the arc length, $\theta$ is the central angle in degrees, and $r$ is the radius of the circle.

Q: What is the central angle in the formula?

A: The central angle is the angle formed by two radii of the circle. It is measured in degrees and is used to determine the length of the arc.

Q: How do I find the central angle?

A: The central angle can be found using a protractor or by measuring the angle between two radii of the circle.

Q: What is the radius in the formula?

A: The radius is the distance from the center of the circle to the edge of the circle. It is measured in units such as feet, meters, or inches.

Q: How do I find the radius?

A: The radius can be found by measuring the distance from the center of the circle to the edge of the circle using a ruler or a measuring tape.

Q: What is the formula for arc length in terms of radians?

A: The formula for arc length in terms of radians is: $s = \theta \cdot r$ , where $s$ is the arc length, $\theta$ is the central angle in radians, and $r$ is the radius of the circle.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the formula: $\theta_{\text{radians}} = \theta_{\text{degrees}} \cdot \frac{\pi}{180}$ .

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

A: The arc length of a circle is a fraction of the circumference of the circle. The fraction is determined by the central angle.

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

A: The circumference of a circle can be found using the formula: $C = 2\pi r$ , where $C$ is the circumference and $r$ is the radius of the circle.

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

A: The arc length of a circle is directly proportional to the central angle. As the central angle increases, the arc length also increases.

Q: How do I use the arc length formula to solve problems?

A: To use the arc length formula to solve problems, you can plug in the values for the central angle, radius, and arc length, and then solve for the unknown value.

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

A: Arc length has many real-world applications, including engineering, physics, and navigation. It is used to calculate the length of curves, the distance traveled by an object, and the angle of a curve.

Q: How do I apply the arc length formula to solve problems in engineering?

A: To apply the arc length formula to solve problems in engineering, you can use it to calculate the length of curves, the distance traveled by an object, and the angle of a curve. You can also use it to design and optimize systems, such as bridges and roads.

Q: How do I apply the arc length formula to solve problems in physics?

A: To apply the arc length formula to solve problems in physics, you can use it to calculate the distance traveled by an object, the angle of a curve, and the length of a curve. You can also use it to model and analyze the motion of objects, such as projectiles and pendulums.

Q: How do I apply the arc length formula to solve problems in navigation?

A: To apply the arc length formula to solve problems in navigation, you can use it to calculate the distance traveled by a ship or an airplane, the angle of a curve, and the length of a curve. You can also use it to design and optimize routes, such as flight paths and shipping routes.