Problem 36 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Find the length of the arc that subtends the given central angle \(\theta\) on a circle of diameter \(d\). (b) Find the area of the sector determined by \(\theta\). \(\theta=2.2, \quad d=120 \mathrm{~cm}\)

Step-by-Step Solution

Answer

Length of arc is 132 cm and area of sector is 3960 cm².

Step 1: Understand Problem Parameters

You are given a circle with diameter \(d = 120\, cm\) and a central angle \(\theta = 2.2\, radians\). We need to find the arc length and the area of the sector corresponding to this angle.

Step 2: Calculate the Radius

The radius of the circle can be found using the formula \(r = \frac{d}{2}\). With \(d = 120\, cm\), the radius is \(r = \frac{120}{2} = 60\, cm\).

Step 3: Formula for Arc Length

The arc length \(L\) can be determined using the formula \(L = r \cdot \theta\).

Step 4: Compute Arc Length

Substitute \(r = 60\, cm\) and \(\theta = 2.2\, radians\) into the arc length formula: \(L = 60 \times 2.2 = 132\, cm\).

Step 5: Formula for Area of Sector

The area \(A\) of a sector is given by \(A = \frac{1}{2} r^2 \theta\).

Step 6: Compute Area of Sector

Substitute \(r = 60\, cm\) and \(\theta = 2.2\, radians\) into the area formula: \(A = \frac{1}{2} \times 60^2 \times 2.2 = 3960\, cm^2\).

Key Concepts

Central Angle

A central angle is a key element when dealing with circles in radial geometry. It is the angle subtended at the center of the circle by two radii. In our example, - The central angle \(\theta\) is given as \(2.2\) radians.- Radians are often used in mathematics because they provide a natural way of measuring angles based on arc lengths.To visualize, imagine the central angle as the 'pie slice' that encompasses part of the circle at its center. The larger the central angle, the larger the corresponding arc length and sector area will be.

Sector Area

The sector area is the region bounded by two radii and the arc between them.- For a circle, the formula to find the sector area \(A\) is \(A = \frac{1}{2} r^2 \theta\).- This indicates that the area is influenced by both the radius and the central angle.In our case, with \(r = 60 \, cm\) and \(\theta = 2.2\) radians:- The calculation becomes \(A = \frac{1}{2} \times 60^2 \times 2.2 = 3960 \, cm^2\).The concept of sector area is crucial when it comes to finding the proportion of a circle occupied by a given angle. It's essentially like determining the size of a portion of pie from the whole circle.

Radial Geometry

Radial geometry focuses on objects and shapes that are related to circles. Understanding radial geometry helps us solve problems involving circles, like finding arc lengths or sector areas. - The circle's radial properties aid in understanding symmetric properties. - Radial lines, or radii, are key components. In practical terms, radial geometry makes it easier to explore how different central angles and radii determine the properties of the circle such as arc length and sector area. Recognizing these radial properties simplifies the overall analysis.

Circle Radius

The radius of a circle is the constant distance from the center of the circle to any point on its boundary. It plays a fundamental role in calculating various properties of the circle.- The radius \(r\) is half of the diameter, so \(r = \frac{d}{2}\).- In our exercise, with a diameter \(d = 120\, cm\), the radius calculates to \(r = 60\, cm\).The radius is not just critical for determining the central properties of a circle. It is also foundational in formulas such as:

• The arc length formula \(L = r \cdot \theta\).
• The sector area formula \(A = \frac{1}{2} r^2 \theta\).

This demonstrates the universal importance of the radius in figuring out dimensions related to circular elements.