Problem 129 - A Graphical Approach to Precalculus with Limits

Question

The unusual corral in the figure is separated into 26 areas, many of which approximate sectors of a circle. Assume that the corral has a diameter of 50 meters. (a) Approximate the central angle for each region, assuming that the 26 regions are all equal sectors with the fences meeting at the center. (b) What is the area of each sector (to the nearest square meter)?

Step-by-Step Solution

Answer

Each sector has a central angle of 13.85 degrees and an area of approximately 60 square meters.

Step 1: Calculate the Total Angle of a Circle

The total angle in a circle is 360 degrees. This is a fundamental concept in geometry that allows us to calculate the angles of individual sectors when they are equal.

Step 2: Determine the Central Angle for Each Sector

To determine the central angle for each sector, divide the total angle of the circle by the number of sectors. Since there are 26 equal sectors, the central angle for each is given by \( \frac{360}{26} \) degrees. Calculating this gives: \[ \text{Central Angle} = \frac{360}{26} \approx 13.85 \text{ degrees} \].

Step 3: Calculate Radius of the Circle

The diameter of the circle is given as 50 meters. The radius \( r \), which is half the diameter, is therefore \( \frac{50}{2} = 25 \) meters.

Step 4: Calculate the Area of Each Sector

The area of a sector of a circle is given by the formula \( A = \frac{\theta}{360} \cdot \pi r^2 \), where \( \theta \) is the central angle in degrees and \( r \) is the radius. Plug in the values: \( \theta = 13.85 \) degrees, \( r = 25 \) meters, to get:\[ A = \frac{13.85}{360} \cdot \pi \times 25^2 \approx 60 \text{ square meters} \].

Key Concepts

Central Angle Calculation

To grasp the concept of calculating a central angle, imagine a pizza being cut into equal slices. The angle at the tip of each slice is akin to the central angle in a circle. In our exercise, the pizza has 26 equal slices, or sectors, in geometry terms.First, recognize that a full circle encompasses 360 degrees. This is a basic but crucial fact about circles that aids in calculating individual sector angles. When these sectors are equal, such as in our exercise with 26 regions, each sector will have the same central angle. To find the central angle of each sector, divide the total circle's degrees by the number of sectors. Here, that process is represented by the formula:

• Central Angle of Each Sector = \( \frac{360}{26} \) degrees

When calculated, each sector's central angle is approximately 13.85 degrees. This simple division gives clarity on how angles distribute evenly among equal portions of a shape like a circle.

Sector Area Formula

Understanding the area of a sector within a circle involves both geometry and a touch of algebra. The sector area formula helps discover how much 'pizza' each slice contains. In our geometric circle, the area of a sector depends on two things:

• The central angle \( \theta \) of the sector
• The radius \( r \) of the circle

The sector area formula looks like this:

• \( A = \frac{\theta}{360} \cdot \pi r^2 \)

This formula essentially carves out a portion of the circle using the given angle. The term \( \pi r^2 \) represents the total area of the circle. Therefore, \( \frac{\theta}{360} \) scales it down to just the desired portion. In our exercise, substituting \( \theta = 13.85 \) degrees and \( r = 25 \) meters into the formula calculates the sector area to be approximately 60 square meters.

Circle Measurements

When working with circles, understanding measurements like diameter and radius is fundamental. These two measurements are closely related. The diameter runs straight through the circle's center from one edge to the other, while the radius is half of that distance.In our example, the circle (or corral) has a diameter of 50 meters. To find the radius, you simply halved the diameter:

• Radius \( r = \frac{50}{2} = 25 \) meters.

These measurements play a key role in other circle calculations, such as perimeter (also known as circumference), which is given by \( 2\pi r \) or \( \pi d \), and area, which is \( \pi r^2 \).
In applying these measurements to the sector area formula, understanding the radius empowers you to accurately calculate not just the portion of the circle, but align it with the dimensions you're working with. Knowing these circle measurements ensures your calculations are grounded in the circle's actual size.