Problem 66 - Precalculus Mathematics for Calculus

Question

A sector of a circle of radius \(24 \mathrm{mi}\) has an area of \(288 \mathrm{mi}^{2}\) Find the central angle of the sector.

Step-by-Step Solution

Answer

The central angle is 1 radian.

Step 1: Understand the given formula

The area of a sector of a circle is given by the formula \( A = \frac{1}{2} r^2 \theta \), where \( A \) is the area, \( r \) is the radius, and \( \theta \) is the central angle in radians. We need this formula to solve for the central angle \( \theta \).

Step 2: Plug in the known values

We are given that the radius \( r = 24 \) miles and the area \( A = 288 \) square miles. Substituting these values into the sector area formula gives us: \[ 288 = \frac{1}{2} (24)^2 \theta \]

Step 3: Simplify the equation

Calculate \( 24^2 \), which is 576. Then our equation becomes: \[ 288 = \frac{1}{2} \times 576 \times \theta \].Divide 576 by 2 to simplify to \[ 288 = 288 \theta \].

Step 4: Solve for \( \theta \)

Divide both sides of the equation by 288 to isolate \( \theta \). This gives us \( \theta = 1 \).

Step 5: Interpret the result

Since \( \theta = 1 \) is in radians, this means the central angle for the sector of the circle is \( 1 \) radian.

Key Concepts

Area of a Sector

The area of a sector is a portion of a circle's total area. Imagine a slice of pie; this sector slice is defined by two radii and the arc between them. Understanding how to calculate this area is crucial for various math applications.

The formula to determine the area of a sector is:

• \( A = \frac{1}{2} r^2 \theta \)

Where:

• \( A \) is the area of the sector
• \( r \) is the radius of the circle
• \( \theta \) is the central angle in radians

This formula shows that the area is directly influenced by the size of the radius and the central angle. A larger radius or a larger central angle results in a larger sector area.

To find the area, insert the values you know into the formula. In problems that provide the area's value and ask for the angle, you simply rearrange the formula to find the missing part, as shown in the original exercise.

Circle Radius

The radius of a circle is the distance from its center to any point along its circumference. It's a fundamental circle measure that plays a key role in all circle-related calculations, including the area of a sector.

The radius influences a circle's size directly — the larger the radius, the larger the circle. Hence, for any sector calculations, a crucial step is knowing or determining the radius accurately.

For example, in the exercise above, the radius is given as 24 miles. Inserting this into any formula regarding the circle or its sector is straightforward. The radius appears in the sector area formula squared, emphasizing its impact on the area.

Circle Sectors

A sector is a section of a circle, similar to a pie slice, that is bounded by two radii and the arc between them. Calculating the properties of sectors, such as their area or arc length, is a common problem in geometry.

Understanding sectors is essential as they form parts of whole circles and can represent real-life situations such as parts of a wheel or slices of pizza.

• The central angle in radians often defines the extent of the sector.
• The area depends significantly on this angle and the circle's radius.

Every sector can be visualized as a fraction of the entire circle. The sector angle is crucial in determining this fraction, further influencing the sector's area or arc length in calculations. By transforming angle units from degrees to radians, one can easily apply these in most mathematical equations regarding sectors.