Problem 56 - Algebra and Trigonometry

Question

A circular arc of length 3 \(\mathrm{ft}\) 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 6.85 ft.

Step 1: Identify the Relationship

To solve this problem, we start by understanding the relationship between the arc length, radius, and central angle. The formula we use is: \[ L = r \theta \] where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the central angle in radians.

Step 2: Convert Angle to Radians

The central angle is given in degrees (\(25^{\circ}\)). We need to convert it to radians using the conversion factor \( \pi/180 \). Thus, the angle in radians \( \theta \) is: \[ \theta = 25^{\circ} \times \frac{\pi}{180} \approx \frac{25\pi}{180} \]}},{

Step 3: Simplify Angle in Radians

Simplify \( \theta = \frac{25\pi}{180} \) to its lowest terms: \[ \theta \approx \frac{5\pi}{36} \]

Step 4: Use Arc Length Formula

Now that we have the angle in radians, substitute \( L = 3 \) ft and \( \theta = \frac{5\pi}{36} \) into the formula \( L = r \theta \): \[ 3 = r \times \frac{5\pi}{36} \]

Step 5: Solve for the Radius

To find \( r \), rearrange the formula to solve for \( r \): \[ r = \frac{3 \times 36}{5\pi} \] Simplifying this expression, we get: \[ r \approx \frac{108}{5\pi} \]

Step 6: Calculate Numerical Value

Finally, calculate the numerical value of \( r \): \[ r \approx \frac{108}{5 \times 3.14159} \approx 6.85 \text{ ft} \]. Therefore, the radius of the circle is approximately 6.85 ft.

Key Concepts

circular arc

A circular arc is a part of the circumference of a circle. Imagine a pie; the edge of one slice is a circular arc. When you take a slice, the crust is the arc, while the slice itself subtends a central angle from the center of the pie (or circle). Circles have many arcs, and by cutting the circle at different angles, you produce different lengths of arcs.

**Key Points about Circular Arcs**

• The length of a circular arc is directly related to both the radius of the circle and the central angle it subtends - think of the arc as the link connecting the radius and angle.
• In this context, finding the arc length requires knowing either the angle and radius or deriving one if the other two are known, as in our exercise.

central angle in radians

The central angle is the angle formed by two radii of a circle. When we measure it, it is usually given in degrees or radians. Radians are particularly useful in mathematical calculations involving circles.

**Understanding Radians**

• Radians provide a natural way to talk about angles, as they relate directly to the circle's radius. One radian is the angle where the arc length equals the radius of the circle.
• Angles in radians make formulas cleaner and simpler, especially in the context of circular motion or calculations involving trigonometric functions.

arc length formula

The arc length formula is an essential tool for problems involving portions of a circle's circumference. It is expressed as:
\[ L = r \theta \] where:

• \( L \) is the arc length, the distance along the curved path.
• \( r \) is the radius, the distance from the center of the circle to any point on its perimeter.
• \( \theta \) is the central angle, measured in radians.

Each variable in the formula is connected, enabling you to solve for the missing value if the other two are known. This formula bridges the spatial (radius) and angular (angle) properties of a circle.

angle conversion degrees to radians

To effectively use the arc length formula, angles must be expressed in radians. But what if the problem gives you degrees? That's where angle conversion comes in.

**How to Convert Degrees to Radians**

• Multiply the angle in degrees by the factor \( \pi/180 \). This factor arises because one complete rotation (360 degrees) is the same as \( 2\pi \) radians.
• When you multiply, you'll convert degree measurement into radians effectively, making calculations possible using our arc length formula.

For example, an angle of \(25^{\circ}\) is converted into radians as follows:
\[\theta = 25^{\circ} \times \frac{\pi}{180} \approx \frac{5\pi}{36}\] This ensures that any subsequent operations using the arc length formula are accurate and mathematically sound.