Problem 10 - Basic Engineering Mathematics

Question

If an angle of \(125^{\circ}\) is subtended by an arc of a circle of radius \(8.4 \mathrm{~cm}\), find the length of (a) the minor arc, and (b) the major arc, correct to 3 significant figures.

Step-by-Step Solution

Answer

Minor arc length is 18.3 cm, and major arc length is 34.5 cm.

Step 1: Understand the Problem

In this problem, we'll be calculating both the minor and major arc lengths of a circle. You are given an angle of \(125^{\circ}\) subtended by an arc and the radius of the circle is \(8.4\,\mathrm{cm}\). The circle's total angle is \(360^{\circ}\). The arc length can be calculated using the formula \(\text{Arc length} = \frac{\theta}{360^{\circ}} \times 2\pi r\). Here, \(\theta\) is the angle in degrees and \(r\) is the radius.

Step 2: Calculate the Minor Arc Length

Use the formula for arc length. For the minor arc, \(\theta = 125^{\circ}\). Plug the values into the formula: \[\text{Minor arc length} = \frac{125}{360} \times 2 \pi \times 8.4 \approx \frac{125}{360} \times 52.8\]\This results in:\[\text{Minor arc length} \approx 18.3 \text{ cm}\] (correct to 3 significant figures).

Step 3: Calculate the Major Arc Length

The major arc is the remainder of the circle after removing the minor arc, so its angle is \(360^{\circ} - 125^{\circ} = 235^{\circ}\). Use the arc length formula again:\[\text{Major arc length} = \frac{235}{360} \times 2 \pi \times 8.4 \approx \frac{235}{360} \times 52.8\]\This results in:\[\text{Major arc length} \approx 34.5 \text{ cm}\] (correct to 3 significant figures).

Step 4: Review and Conclude

To conclude, we have separately calculated the lengths of the minor and major arcs using the given angle and radius. The calculations involve multiplying fractions of the circle's circumference to find each arc length.

Key Concepts

Circle Geometry

Understanding circle geometry is essential when dealing with arc lengths and angles. A circle is a perfectly round shape where all points are equidistant from a central point, called the center. This distance is the radius. The perimeter of a circle is known as the circumference. Circle geometry involves concepts such as:

• Diameter - twice the radius
• Radius - half of the diameter
• Circumference - the total distance around the circle, calculated by the formula \( C = 2\pi r \)
• Arc - a portion of the circumference

Circle geometry is foundational for calculating lengths of arcs as it sets the formulas and principles used.

Angle Subtension

The concept of angle subtension is crucial when finding arc lengths. An angle subtended by an arc is the angle formed at the circle's center by two radii drawn from each endpoint of the arc. In this exercise:

• The given angle subtending the arc is \(125^{\circ}\)
• This is called the minor angle since it is less than \(180^{\circ}\)
• The major angle can be calculated by subtracting the minor angle from \(360^{\circ}\), which is the total angle in a circle

This understanding helps in calculating both the minor and major arc lengths accurately.

Radius

The radius of a circle is the distance from the center point to any point on the circumference. It's a key component in all circle geometry calculations, including finding arc lengths. The problem specifies:

• The circle's radius is \(8.4\, \mathrm{cm}\)

Knowing the radius allows us to determine the circumference using the formula \( C = 2\pi r \). The radius is then used in the arc length formula: \( \text{Arc Length} = \frac{\theta}{360^{\circ}} \times 2\pi r \) to find the proportional segment of the circumference."

Significant Figures

When providing answers, it's important to consider the number of significant figures. Significant figures are digits that carry meaning contributing to the precision of a quantity. For instance:

• The calculated minor arc length is approximately \(18.3\) cm
• The major arc length is approximately \(34.5\) cm

Both answers are rounded to three significant figures, as required by the problem. This ensures that the answers are both precise and practical in terms of significant digits.