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

Step-by-Step Solution

Answer

The arc length is approximately 6.98 meters, and the area of the sector is approximately 27.92 square meters.

Step 1: Convert the Central Angle from Degrees to Radians

The formula for the length of an arc requires the angle in radians. Convert the central angle \(\theta\) from degrees to radians using the conversion formula: \(\text{radians} = \theta \times \left(\frac{\pi}{180}\right)\).Given \(\theta = 50^{\circ}\), the angle in radians is:\[\text{radians} = 50 \times \left(\frac{\pi}{180}\right) = \frac{50\pi}{180} = \frac{5\pi}{18} \text{ radians}\]

Step 2: Calculate the Radius of the Circle

The diameter \(d\) is given as 16 meters. The radius \(r\) is half of the diameter.\[r = \frac{d}{2} = \frac{16}{2} = 8 \text{ meters}\]

Step 3: Calculate the Length of the Arc

The length of the arc \(L\) is given by the formula: \(L = r \times \theta \) (where \(\theta\) is in radians).Using \(r = 8\) meters and \(\theta = \frac{5\pi}{18}\) radians:\[L = 8 \times \frac{5\pi}{18} = \frac{40\pi}{18} = \frac{20\pi}{9} \text{ meters} \approx 6.98 \text{ meters}\]

Step 4: Find the Area of the Sector

The area of the sector \(A\) of the circle is given by:\[ A = \frac{1}{2} \times r^2 \times \theta \]Using \(r = 8\) meters and \(\theta = \frac{5\pi}{18}\) radians:\[A = \frac{1}{2} \times 8^2 \times \frac{5\pi}{18} = \frac{1}{2} \times 64 \times \frac{5\pi}{18} = \frac{320\pi}{36} = \frac{160\pi}{18} = \frac{80\pi}{9} \text{ square meters} \approx 27.92 \text{ square meters}\]

Key Concepts

Understanding Arc Length

In circle geometry, one of the key measurements we encounter is the arc length. An arc is simply a portion of the circumference of a circle. The formula to determine the arc length involves the circle's radius and the central angle in radians. The formula is given by \( L = r \times \theta \), where \( L \) is the arc length, \( r \) is the radius of the circle, and \( \theta \) is the central angle in radians.

Consider the circle with a diameter of 16 meters. Since the radius is half the diameter, you have \( r = 8 \) meters. After converting the 50 degrees to radians (which we'll discuss more below), you use the formula: \( L = 8 \times \frac{5\pi}{18} \). This calculation gives an arc length of approximately 6.98 meters.

This method of finding the arc length is important because it provides a way to measure a circular path without having to traverse it directly. This is particularly useful in fields like engineering and architecture.

Exploring Sector Area

The sector of a circle is the region enclosed by two radii and an arc. Calculating the area of a sector helps understand the portion of the circle that is taken up by that sector. The formula used for finding the area utilizes the radius and the central angle in radians: \( A = \frac{1}{2} \times r^2 \times \theta \).

In this example, the radius \( r \) of 8 meters and the central angle in radians \( \theta = \frac{5\pi}{18} \) are used in the formula, resulting in an area of \( A = \frac{1}{2} \times 8^2 \times \frac{5\pi}{18} \), or approximately 27.92 square meters.

Understanding how to calculate the sector area is vital in practical applications like determining the space used in a circular segment or designing a circular layout. It allows you to visualize and quantify curved space precisely.

Mastering Radian Conversion

Before performing calculations involving angles in a circle, angles often need conversion from degrees to radians. This conversion is crucial due to how formulas in circle geometry are derived using radians. The relationship between degrees and radians is given by the formula: \( \text{radians} = \theta \times \left(\frac{\pi}{180}\right) \), where \( \theta \) is the angle in degrees.

Using this conversion for 50 degrees gives: \( \text{radians} = 50 \times \frac{\pi}{180} = \frac{5\pi}{18} \) radians. By understanding conversion, you ensure your calculations for arc length and sector area are reliable and accurate.

Utilizing radian conversion is crucial not only in math but in science and engineering, where precise measurements and calculations are necessary. Therefore, mastering this concept ensures a solid foundation in solving problems involving circular metrics.