Problem 58 - Precalculus Mathematics for Calculus

Question

A circular arc of length 3 ft subtends a central angle of \(25^{\circ}\). Find the radius of the circle.

Step-by-Step Solution

Answer

The radius of the circle is approximately 6.876 ft.

Step 1: Understand the Relationship Between Arc Length, Radius, and Angle

An arc length \(s\), a radius \(r\), and a central angle \(\theta\) (in radians) are related by the formula: \[ s = r\theta \]. First, we need to convert the angle from degrees to radians.

Step 2: Convert Degrees to Radians

The formula to convert degrees to radians is given by \( \theta_{\text{radians}} = \frac{\pi}{180} \times \theta_{\text{degrees}} \). For our angle, \( \theta = 25^{\circ} \), it becomes: \[ \theta = \frac{\pi}{180} \times 25 = \frac{25\pi}{180} = \frac{5\pi}{36} \text{ radians} \].

Step 3: Solve for the Radius

Now that we have \( \theta \) in radians, we can use the formula from Step 1. Rearrange the equation \( s = r\theta \) to solve for \( r \): \[ r = \frac{s}{\theta} \]. Substitute \( s = 3 \) ft and \( \theta = \frac{5\pi}{36} \text{ radians} \): \[ r = \frac{3}{\frac{5\pi}{36}} = \frac{3 \times 36}{5\pi} = \frac{108}{5\pi} \].

Step 4: Calculate the Radius

Perform the actual calculation to find \( r \): \[ r \approx \frac{108}{5\times3.14159} \approx \frac{108}{15.70795} \approx 6.876 \text{ ft} \]. This is the radius of the circle.

Key Concepts

Arc Length

An arc length is simply a portion of the circumference of a circle. Imagine cutting out a small slice of a pie – the crust of that slice represents the arc length. Arc length is primarily dependent on two factors: the radius of the circle and the central angle that this arc spans. The larger the radius or the more extensive the angle, the longer the arc length will be. The formula used to calculate this is

• \( s = r\theta \)

where \( s \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians. Understanding this formula is crucial when trying to determine one unknown value, given the other two.

Radians Conversion

Radians are a way of measuring angles based on the radius of a circle. Unlike degrees, which divide a circle into 360 parts, radians measure angles using the circle's radius. One complete revolution around a circle is \( 2\pi \) radians, which equals 360 degrees. Converting from degrees to radians is pivotal in many geometry problems, like the one we're solving. To convert degrees to radians, you multiply the angle in degrees by \( \frac{\pi}{180} \). So, for an angle of \( 25^{\circ} \), we perform the calculation:

• \( \theta_{\text{radians}} = \frac{\pi}{180} \times 25 = \frac{5\pi}{36} \)

This conversion is essential when working with formulas that require angles to be in radians rather than degrees.

Central Angle

The central angle is the angle formed by two radii extending from the center of a circle to its circumference. It's called 'central' because its vertex is at the circle's center. The size of this angle determines the arc length for any given radius. In our example, the central angle is \(25^{\circ}\), which we converted to radians to facilitate calculations involving arc length. Central angles are integral in problems involving arc length because they directly influence how much of the circumference is being measured. Knowing the central angle's measure in radians is necessary to use the basic arc length formula effectively.

Radius Calculation

Calculating the radius from a known arc length and central angle involves algebraically rearranging the arc length formula. In simpler terms, if you know how long the arc is and the angle it subtends, you can find the radius. Starting from the formula

• \( s = r\theta \)

we rearrange it to solve for \( r \), the radius:

• \( r = \frac{s}{\theta} \)

With an arc length \( s = 3 \) ft and a converted angle \( \theta = \frac{5\pi}{36} \) radians, plugging into the formula gives:

• \( r = \frac{3}{\frac{5\pi}{36}} = \frac{108}{5\pi} \)

Through numerical calculation, this simplifies to approximately 6.876 feet. This method of finding the radius is straightforward once you have the necessary pieces of information.