Problem 64 - Algebra and Trigonometry

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 Problem

The problem gives us the area of a sector of a circle and the radius of the circle. We need to find the central angle of the sector in radians.

Step 2: Recall the Formula for the Area of a Sector

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

Step 3: Substitute Known Values into the Formula

We have the following values given from the problem: \( A = 288 \, \text{mi}^2 \) and \( r = 24 \, \text{mi} \). Substitute these into the formula to get \[ 288 = \frac{1}{2} \times 24^2 \times \theta \].

Step 4: Solve for the Central Angle \( \theta \)

First, calculate \( 24^2 = 576 \). Then substitute in to solve for \( \theta \): \[ 288 = \frac{1}{2} \times 576 \times \theta \].\ Simplifying gives \[ 288 = 288 \times \theta \]. Thus, \( \theta = 1 \) radian.

Key Concepts

Central Angle

The central angle is the angle whose vertex is at the center of a circle and whose sides extend to the circumference, slicing the circle into a sector. Imagine a pizza, where the central angle is like the angle formed at the tip of a slice pointing to the center of the pizza.
In the context of a circle, the central angle is crucial in determining the size of the corresponding arc and the area of the sector. For example, if the whole circle makes a 360-degree angle, and the central angle is a smaller fraction of it, then the sector size is similarly reduced.

• The central angle is often measured in radians, which is a unit more suitable for mathematical calculations.
• Radians provide a more direct relationship between the circle's radius and the arc length.

In our exercise, calculating the central angle helps in understanding the size of our sector in comparison to the entire circle.

Area of a Sector

The area of a sector is like a piece of the pie from the whole circular pie. When you cut a circle into pieces, each piece or 'sector' has an area. This area depends on two things: how big the piece is (measured by the central angle) and how large the circle is (measured by the radius).
The formula for the area of a sector is given as \( A = \frac{1}{2} r^2 \theta \), where:

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

This relationship shows how the area grows with the square of the radius and the size of the central angle in radians.
In our given problem, knowing the area of the sector and radius allows us to backtrack and find the missing central angle.

Radians

Radians are a way of measuring angles based on the radius of the circle. Unlike degrees, which divide a circle into 360 parts, radians consider the circle's circumference. One radian is the angle made when the radius is wrapped around the circle's edge.
Why use radians? They are incredibly useful in calculus and physics because they simplify many formulas by directly linking angle measures to the arc length and radius.

• For a complete circle, there are \(2\pi\) radians, which equates to 360 degrees.
• A half-circle or semi-circle measures \(\pi\) radians.
• One radian is approximately 57.2958 degrees.

In problems like ours, converting between degrees and radians is common but knowing the definition helps when directly calculating central angles in radians.