Problem 30 - Algebra and Trigonometry with Analytic Geometry

Question

If a circular arc of the given length \(s\) subtends the central angle \(\boldsymbol{\theta}\) on a circle, find the radius of the circle. \(s=3 \mathrm{~km}, \quad \theta=20^{\circ}\)

Step-by-Step Solution

Answer

The radius of the circle is \( \frac{27}{\pi} \) km.

Step 1: Convert Angle to Radians

First, convert the given angle \( \theta \) from degrees to radians, since arc length formulas typically use radians. The conversion formula is \( \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \). Thus, for \( \theta = 20^{\circ} \), the conversion gives \[ \theta_{\text{radians}} = 20 \times \frac{\pi}{180} = \frac{\pi}{9} \, \text{radians}. \]

Step 2: Use Arc Length Formula

Using the arc length formula \( s = r \theta \) where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians, we can solve for \( r \). We have \( s = 3 \) km and \( \theta = \frac{\pi}{9} \) radians. So, \[ 3 = r \times \frac{\pi}{9}. \]

Step 3: Solve for the Radius

Rearrange the formula to solve for the radius \( r \): \[ r = \frac{3}{\frac{\pi}{9}} = 3 \times \frac{9}{\pi} = \frac{27}{\pi}. \] Hence, the radius of the circle is \( \frac{27}{\pi} \) km.

Key Concepts

Arc Length

The arc length of a circle is a measure of the distance along a section of the circle's circumference. This distance can be thought of as the circle's 'curved edge' from one point to another along its outer line. Think of it as the part of the circumference that looks like a slice of a pizza or a pie.Arc length is mathematically given by the formula:\[ s = r \theta \]where:

• \( s \) is the arc length,
• \( r \) is the radius of the circle, and
• \( \theta \) is the central angle in radians.

Using this formula, we can determine any one of those variables, provided the other two are known. In our exercise, we used this formula to solve for the radius \( r \). Knowing the arc length and the angle, the task was to rearrange the equation and isolate the radius.

Central Angle

The central angle is the angle subtended at the center of the circle by an arc. If you imagine the center of the circle as the point from which your fingers spread out to touch the ends of the arc, the central angle is the angle made by your fingers. Measuring the central angle is crucial because it determines the fraction of the circle's circumference that your arc length occupies.
In this context, a full circle's central angle is \(360^{\circ}\) or \(2\pi\) radians.When solving problems like our exercise, it's vital to use central angles to find the relationship between arc length and radius. Knowing the central angle helps in using the arc length formula effectively, pegged on understanding this segment of the circle.

Angle Conversion

For many mathematical calculations involving circles, angles are more conveniently expressed in radians rather than degrees. It is often necessary to convert angles from degrees to radians. To convert an angle from degrees \( \theta_{\text{degrees}} \) to radians \( \theta_{\text{radians}} \), you use the formula:\[ \theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180} \]This conversion is essential in calculations involving trigonometry and calculus, where the natural constant \( \pi \) helps maintain accuracy in computations.
In the given exercise, the angle \(20^{\circ}\) has been converted to radians as \(\frac{\pi}{9}\), which allows the arc length formula to be used effectively.

Radians

Radians provide a natural and mathematical way to describe angles based on the radius of a circle. In simple terms, a radian is the angle made when the arc length equals the radius of the circle.A full circle has \(2\pi\) radians, which equates to \(360^{\circ}\). Therefore, the relationship between radians and degrees is straightforward:

• \( \pi \) radians equals \(180^{\circ} \).
• \(1\) radian equals approximately \(57.2958^{\circ}\).

Understanding radians is vital for solving problems involving circles because it simplifies mathematics related to circular motion and waves.
By expressing angles in radians, functions like sine and cosine become simpler and more intuitive. In our context, using radians permitted seamless calculation of the circle's radius using arc length.