Problem 75 - Precalculus with Calculus Previews

Question

In Problems \(73-76\), find the area of the circular sector having the given radius \(r\) and central angle \(\theta\). $$ r=6 \mathrm{~m}, \theta=30^{\circ} $$

Step-by-Step Solution

Answer

The area of the sector is \(3\pi\) m\(^2\).

Step 1: Understand the Problem

We need to find the area of a circular sector. The problem provides the radius \( r = 6 \text{ m} \) and the central angle \( \theta = 30^{\circ} \). A circular sector is a portion of a circle bounded by two radii and an arc.

Step 2: Convert Angle to Radians

The formula for the area of a circular sector involves the central angle in radians. First, convert \( \theta = 30^{\circ} \) into radians. Use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). Thus, \( 30^{\circ} = 30 \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}. \)

Step 3: Use the Sector Area Formula

The area \( A \) of a circular sector is given by the formula \( A = \frac{1}{2} r^2 \theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. Substitute \( r = 6 \text{ m} \) and \( \theta = \frac{\pi}{6} \text{ radians} \) into the formula.

Step 4: Calculate the Area

Insert the values into the formula: \( A = \frac{1}{2} \times 6^2 \times \frac{\pi}{6} \). Calculate \( 6^2 = 36 \), and then compute \( \frac{1}{2} \times 36 \times \frac{\pi}{6} = \frac{36}{12} \pi = 3\pi \).

Step 5: Provide Final Answer

The area of the circular sector is \( 3\pi \) square meters. If a decimal approximation is needed, \( 3\pi \approx 9.42 \text{ m}^2 \).

Key Concepts

Central Angle Conversion

To solve problems involving circular sectors, we first need to express the central angle in the correct units. Typically, the central angle might be given in degrees. However, for mathematical calculations involving sectors, radians are preferred. This is because many formulas, including that for sector area, are simpler in radians.
Converting degrees to radians is straightforward. Use the conversion formula:

• Radians = Degrees \( \times \frac{\pi}{180} \)

For example, if you have an angle of \( 30^{\circ} \), you convert it to radians as follows:

• \( 30^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6} \) radians

Converting to radians streamlines further calculations.

Radian Measure

The radian is an alternative to degrees for measuring angles. It provides a more natural way of expressing angles in mathematics. One radian is defined as the angle made by taking the radius of a circle and wrapping it along the circle’s edge.
Knowing how to convert between degrees and radians simplifies many calculations in trigonometry and calculus. The full circle in radians is \( 2\pi \), equivalent to \( 360^{\circ} \). Thus, \( \pi \) radians equals \( 180^{\circ} \), helping us understand the conversion factor of \( \frac{\pi}{180} \). This relationship is crucial when dealing with angular measurements in various mathematical problems.

Sector Area Formula

The formula for finding the area of a circular sector is particularly neat and derived specifically when the central angle is in radians. Given a radius \( r \) and a central angle \( \theta \) in radians, the area \( A \) of a sector is:

• \( A = \frac{1}{2} r^2 \theta \)

This formula reflects the proportionality between the angle and the circle's whole. It essentially calculates what fraction of the circle's total area is represented by the sector, based on the angle. It simplifies the calculation by taking advantage of the sector being a simple fraction \( \theta \) over the total \( 2\pi \). Understand that this makes it direct to plug and play the problem’s data to find the solution.

Geometry Problem Solving

Approaching geometry problems such as finding the area of a circular sector involves a systematic process:

• First, understand the problem and what needs solving.
• Identify all given values, such as radius and angle.
• Ensure all measurements are in compatible units for your formula.
• If necessary, convert units, such as degrees to radians.
• Apply the appropriate formula to find your solution.

Being systematic ensures that you tackle each element of the problem methodically, reducing errors and confusion. Ensuring your given data is correctly recalibrated to suit formulaic requirements can notably enhance accuracy.

Unit Conversion

Unit conversion is crucial in mathematics to ensure consistency and accuracy. When solving problems involving different types of measurements or calculations, matching units is crucial.
For example, the radius might be given in meters while the angle in degrees. Before using the sector area formula, ensure the central angle is converted into radians. Unit conversion often involves:

• Degrees to radians (as detailed earlier)
• Converting units of length if necessary, though in many cases, like meters, it's direct

Maintaining consistent units throughout a calculation avoids errors and ensures the process flows smoothly, providing accurate results.