Problem 73 - Technical Mathematics with Calculus

Question

Find the area of a sector having a radius of 5.92 in. and a central angle of \(62.5^{\circ} .\)

Step-by-Step Solution

Answer

The area of the sector is approximately 19.1 square inches.

Step 1: Understand the concept of a sector area

The area of a sector of a circle can be found using the formula: Area of a sector = \(\frac{n}{360} \times \pi \times r^2\), where 'n' is the central angle in degrees, 'r' is the radius of the circle, and \(\pi\) is a constant approximately equal to 3.14159.

Step 2: Plug in the given values

In the given exercise, the radius \(r\) is 5.92 inches and the central angle \(n\) is \(62.5^\circ\). To find the area of the sector, substitute those values into the formula: Area of a sector = \(\frac{62.5}{360} \times \pi \times (5.92)^2\).

Step 3: Calculate the area of the sector

Perform the calculation: \(\frac{62.5}{360} \approx 0.17361\) (rounded off to five decimal places), and \((5.92)^2 = 35.0464\) (rounded off to four decimal places). Then, multiply these two values with \(\pi\): Area of a sector = \(0.17361 \times \pi \times 35.0464\). The use of a calculator will give the numeric result.

Step 4: Obtain the final result

After calculation, the area of the sector to an appropriate number of significant figures (depending on your instruction for rounding) will be: Area ≈ 19.1 square inches.

Key Concepts

Central Angle

Understanding the central angle is fundamental when studying the properties of a circle. The central angle is formed by two radii extending from the center of the circle to the circumference, creating an arc on the circle's edge. This angle is a slice of the 360 degrees that make up the entire circle. In the case of our exercise, the central angle is explicitly given as 62.5 degrees. This measurement is crucial because it represents the proportion of the circle the sector occupies. A larger central angle would result in a larger sector, and conversely, a smaller angle would yield a smaller sector.

Radius

The radius is a straight line from the center of a circle to any point on its circumference. It also happens to be half the length of the circle's diameter. In this exercise, the radius of the sector is 5.92 inches. The radius is not just a measurement; it's the backbone of many calculations in circle geometry, including the area of a circle, circumference, and in our case, the area of a sector. All points on the circumference are equidistant from the center, which is the definition of a circular shape, and that's why all radii of a circle will be the same length.

Sector Area Calculation

The calculation for the area of a sector requires an understanding of both the central angle and the radius you've just learned. The sector area is a fraction of the circle's total area dependent on the central angle. The formula to calculate it is Area of sector = (central angle/360) x π x radius squared. In the exercise, we apply this formula by substituting the given central angle and radius into it. The area is proportionate to the angle, as the angle's ratio to 360 degrees determines what fraction of the total circle area the sector represents. The inclusion of π, approximately 3.14159, is vital as it represents the ratio of the circumference of any circle to its diameter, which is a constant for all circles. Thus, the formula encompasses all the essential aspects of a circle's geometry.

Pi (π)

Pi, denoted by the Greek letter π, is one of the most well-known mathematical constants. It represents the ratio of the circumference of any circle to its diameter, and it's approximately equal to 3.14159. Pi is irrational, meaning it has an infinite number of non-repeating decimals. This concept is vital to calculate the area of a circle and by extension, the area of a sector of a circle. In our formula, π is multiplied by the square of the radius, and this product gives us the area of a whole circle. When we multiply this by the fraction of the central angle over 360 degrees, we get the proportion of that whole circle that our sector represents.