Problem 35 - 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 \(\theta\) $$\theta=50^{\circ}, \quad d=16 \mathrm{m}$$

Step-by-Step Solution

Answer

Arc length: \(\frac{20\pi}{9} \, \text{m}\); Sector area: \(\frac{80\pi}{9} \, \text{m}^2\).

Step 1: Convert the angle from degrees to radians

To find the length of an arc and the area of a sector, angle \(\theta\) needs to be in radians. Use the conversion formula, \(1^{\circ} = \frac{\pi}{180} \, \text{radians}\).\[\theta = 50^{\circ} \times \frac{\pi}{180} = \frac{5\pi}{18} \, \text{radians}\]

Step 2: Calculate the radius of the circle

Given that the diameter \(d\) of the circle is \(16 \, \text{m}\), the radius \(r\) is half of the diameter.\[r = \frac{16}{2} = 8 \, \text{m}\]

Step 3: Calculate the length of the arc

The formula for the arc length \(L\) is given by \(L = \theta \times r\), where \(\theta\) is in radians and \(r\) is the radius.\[L = \frac{5\pi}{18} \times 8 = \frac{40\pi}{18} = \frac{20\pi}{9} \, \text{m}\]

Step 4: Calculate the area of the sector

The formula for the area \(A\) of a sector is \(A = \frac{1}{2} \times r^2 \times \theta\).\[A = \frac{1}{2} \times 8^2 \times \frac{5\pi}{18} = \frac{1}{2} \times 64 \times \frac{5\pi}{18} = \frac{160\pi}{18} = \frac{80\pi}{9} \, \text{m}^2\]

Key Concepts

Arc Length

The arc length of a circle is a portion of the circumference, shaped by the path that a point travels along the perimeter. It acts as the curved segment of the circle, which is defined by a central angle. Think of it as a slice of the edge of a pie. To compute the arc length, you need two crucial things: the radius of the circle and the central angle in radians.
The formula for arc length is given by:

• \( L = \theta \times r \)

where \( L \) is the arc length, \( \theta \) is the central angle in radians, and \( r \) is the radius of the circle.
For instance, if you have a circle with a radius of 8 meters and an angle of \( \frac{5\pi}{18} \) radians, to find the arc length, simply multiply these two values. This reflects the actual distance along the curved portion of the circle, making it an important measurement in various calculations and applications.

Radian Conversion

Radians are the standard measurement for angles in mathematics, particularly when dealing with circles. Unlike degrees, which divide a circle into 360 equal parts, radians provide a natural way of expressing angles based on the circle's radius. One full circle is equal to \( 2\pi \) radians. To transition from degrees to radians, use the conversion factor:

• \(1^\circ = \frac{\pi}{180} \text{ radians} \).

Let's convert an angle of 50 degrees into radians. Multiply the degree measure by \( \frac{\pi}{180} \). Thus, \(\theta = 50^\circ \times \frac{\pi}{180} = \frac{5\pi}{18} \) radians.
This conversion is crucial as many formulas, including those for arc length and areas of sectors, require the angle to be in radians. Ensuring your angles are in the correct format helps maintain accuracy in mathematical operations.

Central Angle

The central angle is pivotal when exploring circle geometry. It's the angle with its vertex at the center of the circle and whose sides (called radii) extend to the circle's perimeter. This angle dictates the size of the arc and the corresponding sector of the circle.
When dealing with a circle, the central angle determines specific properties, such as:

• Arc Length - size of the angle affects the length of the arc.
• Area of Sector - dictates the portion of the circle's area encapsulated by the arc.

Typically measured in radians, formulas involving the central angle become straightforward. For instance, calculating the arc length or the area of a sector involves multiplying the radian measure of the central angle by the radius or squared radius, respectively. The central angle is an essential player in understanding and manipulating circle properties.