Problem 64 - Precalculus Mathematics for Calculus

Question

A sector of a circle has a central angle of \(60^{\circ} .\) Find the area of the sector if the radius of the circle is \(3 \mathrm{mi}\).

Step-by-Step Solution

Answer

The area of the sector is \( \frac{3}{2} \pi \) square miles.

Step 1: Understand the formula for the area of a sector

The formula to calculate the area of a sector with central angle \( \theta \) (in degrees) and radius \( r \) is given by \( \text{Area} = \frac{\theta}{360} \times \pi r^2 \). This formula is derived from the proportion of the angle of the sector to the whole angle of the circle, which is \(360^{\circ}\).

Step 2: Plug in the given values

We are given that the central angle \( \theta \) is \( 60^{\circ} \) and the radius \( r \) is \( 3 \) miles. Substitute these values into the formula to find the area of the sector: \[ \text{Area} = \frac{60}{360} \times \pi \times 3^2 \].

Step 3: Simplify the fraction

Simplify the fraction \( \frac{60}{360} \). Divide both the numerator and the denominator by their greatest common divisor, which is \(60\). This gives \( \frac{60}{360} = \frac{1}{6} \).

Step 4: Calculate the area of the sector

Now that we have \( \frac{1}{6} \), substitute it back into the equation: \[ \text{Area} = \frac{1}{6} \times \pi \times 3^2 \]. Calculate \(3^2\), which is \(9\), thus \[ \text{Area} = \frac{1}{6} \times \pi \times 9 \].

Step 5: Final calculation

Multiply the fractions: \( \frac{1}{6} \times 9 = \frac{9}{6} \). Simplify \( \frac{9}{6} \) to \( \frac{3}{2} \). Thus the area is \[ \text{Area} = \frac{3}{2} \pi \].

Key Concepts

Circle Geometry

Circle geometry is a fascinating part of mathematics that deals specifically with shapes and properties related to circles. It is fundamental in understanding the concepts of sectors, which are pie-shaped parts of a circle. A circle itself is a special figure, featuring:

• Symmetry - Every circle is symmetric around its center.
• Constant radius - All points on the edge are equidistant from the center.

In circle geometry, it's important to know that the total angle around the center of a complete circle is \(360^{\circ}\). This property is key when calculating areas and understanding other geometric relationships within a circle. When dealing with problems like finding the area of a sector, being comfortable with circle properties helps simplify the problem.

Central Angle

A central angle is the angle formed when two radii of a circle meet at the center. It's an essential element of circle geometry applied frequently in calculations involving sectors. Here's why central angles are important:

• Central angles help determine the size of the sector they "carve out" in the circle.
• They are measured in degrees, allowing us to express parts of the circle as fractions of \(360^{\circ}\).

In our exercise, the given central angle is \(60^{\circ}\), which is a sixth of the entire circle (because \( \frac{60}{360} = \frac{1}{6} \)). This fraction is precisely why our formula for the area of a sector includes dividing the central angle by \(360\). It's a straightforward way to find how big the sector is in relation to the whole circle.

Radius

The radius is a line segment from the center of a circle to any point on its circumference. It is a crucial measurement because:

• The radius is used to calculate the area, circumference, and other properties of a circle.
• All radii in a circle are the same length, demonstrating the uniform distance from the center to the circle's edge.

In the case of our exercise, with a radius of \(3\) miles, we see how the radius influences the overall size of the circle and thus the size of any sector. The formula to find the area of a sector involves the square of the radius \( (\pi r^2) \) because it scales directly with the size of both the circle and sector. This means that even a small change in the radius can significantly impact the area of the sector. Understanding the role of the radius in circular equations is crucial for accurate calculations.