Problem 36 - Precalculus: Functions and Graphs

Question

Exer. \(35-36:\) (a) Find the length of the are that subtends the given central angle \(\boldsymbol{\theta}\) 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

The arc length is 132 cm, and the sector area is 3960 cm².

Step 1: Calculate the Radius

To find the radius of the circle, we first need to remember that the radius is half of the diameter. So, if the diameter is \(d = 120\) cm, the radius \(r\) can be calculated as follows:\[ r = \frac{d}{2} = \frac{120}{2} = 60 \text{ cm} \]

Step 2: Find the Arc Length

The formula for the length of an arc, \(L\), is given by \(L = \theta \times r\), where \(r\) is the radius and \(\theta\) is the angle in radians. Substituting the given values, \(\theta = 2.2\) and \(r = 60\) cm, we get:\[ L = 2.2 \times 60 = 132 \text{ cm} \]

Step 3: Calculate the Area of the Sector

The formula for the area of a sector, \(A\), is \(A = \frac{1}{2} \times r^2 \times \theta\). With \(r = 60\) cm and \(\theta = 2.2\), we substitute:\[A = \frac{1}{2} \times (60)^2 \times 2.2 = \frac{1}{2} \times 3600 \times 2.2 = 3960 \text{ cm}^2\]

Key Concepts

Arc Length

Arc length in circle geometry is the distance along the curved line making up the arc. Think of it as the perimeter or boundary of a section of the circle's circumference. To understand how to find the arc length, consider these key points:

• An arc is defined by a central angle, denoted as \(\theta\), and a radius \(r\).
• The formula to calculate the arc length \(L\) is \(L = \theta \times r\), where \(\theta\) must be in radians.
• The greater the angle \(\theta\), the longer the arc length since a larger angle covers more of the circle's circumference.

For example, if you have an angle \(\theta = 2.2\) radians and a radius \(r = 60\) cm, you can easily find the arc length by multiplying these two numbers together, giving you an arc length of 132 cm in this case. Understanding how to derive this using the radius and central angle allows you to solve arc length problems effectively.

Sector Area

The sector area of a circle is the region bounded by two radii and the corresponding arc. Similar to slicing a pie, it's like a wedge from the whole circle.

• The formula to find the area of a sector \(A\) is \(A = \frac{1}{2} \times r^2 \times \theta\).
• This formula accounts for the size of the radius \(r\) and the angle \(\theta\) in radians, because both impact the size of the sector.
• A greater radius or a larger angle increases the area since more of the circle is encompassed.

Let's take an example where \(r = 60\) cm and \(\theta = 2.2\) radians. Plug these into the formula: \[ A = \frac{1}{2} \times (60)^2 \times 2.2 = 3960 \, \text{cm}^2 \]
This result shows how much of the circle's total area the sector occupies, giving a precise measurement of the sector's size.

Radians

Radians are a crucial unit for measuring angles in mathematics, especially in circle geometry. Unlike degrees, which divide a circle into 360 parts, radians give a more natural measure based on the circle's properties.

• A radian is the angle formed when the arc length equals the radius of the circle.
• The circumference of a circle is \(2\pi\) times the radius, meaning there are \(2\pi\) radians in a full circle.
• As a conversion, \(180^{\circ}\) is equivalent to \(\pi\) radians. To convert degrees to radians, multiply by \(\frac{\pi}{180}\).

By using radians, calculations in circle geometry become simplified, as formulas for arc length and sector area rely on \(\theta\) being in radians. For example, when given \(\theta = 2.2\), it's understood directly in terms of the circle's dimensions without needing further conversion.