Problem 3 - Thomas Calculus

Question

You want to make an \(80^{\circ}\) angle by marking an arc on the perimeter of a 12 -in.-diameter disk and drawing lines from the ends of the arc to the disk's center. To the nearest tenth of an inch, how long should the arc be?

Step-by-Step Solution

Answer

The arc length is approximately 8.4 inches.

Step 1: Understand the Circle Geometry

Given a circle with a diameter of 12 inches, we recognize that the radius \( r \) is half of the diameter. So, \( r = \frac{12}{2} = 6 \) inches. We need to calculate the arc length for an angle of \( 80^{\circ} \).

Step 2: Understand Arc Length Formula

The arc length \( L \) for an angle \( \theta \) in a circle of radius \( r \) is given by the formula \( L = r \theta \), where \( \theta \) is measured in radians. First, convert the angle to radians since our angle is given in degrees.

Step 3: Convert Angle from Degrees to Radians

To convert \( 80^{\circ} \) to radians, use the conversion formula: \( 1^{\circ} = \frac{\pi}{180} \). Thus, \( 80^{\circ} = 80 \times \frac{\pi}{180} = \frac{8\pi}{18} = \frac{4\pi}{9} \text{ radians} \).

Step 4: Calculate the Arc Length

Now that we have \( r = 6 \) inches and \( \theta = \frac{4\pi}{9} \) radians, find the arc length \( L \) using the formula \( L = r \theta \). Substitute the values: \( L = 6 \cdot \frac{4\pi}{9} = \frac{24\pi}{9} = \frac{8\pi}{3} \).

Step 5: Simplify and Round the Arc Length

Compute \( \frac{8\pi}{3} \) to get a numerical approximation. Using the value of \( \pi \approx 3.1416 \), we find \( L = \frac{8 \times 3.1416}{3} \approx 8.377 \text{ inches} \). Round this to the nearest tenth: \( L \approx 8.4 \) inches.

Key Concepts

Arc Length

The concept of arc length refers to the distance along the curved path of a circle's perimeter that is defined by a specific angle. In simpler terms, if you imagine walking along just a part of the circumference of a circle, the arc length is how far you would walk.
To calculate arc length, you need two key pieces of information: the radius of the circle and the angle that your arc spans, expressed in radians. This is important because the formula for arc length is:

• \[ L = r \theta \]
• where \( L \) is the arc length, \( r \) is the radius, and \( \theta \) is the angle in radians.

When dealing with circles in real-world applications, finding the arc length can be essential for measuring sectors or designing circular paths. For example, in this exercise, you calculated how long an arc would be on a 12-inch diameter disc for an 80-degree angle. This allows precise marking and cutting, ensuring accuracy in projects that involve arcs.

Radians

Radians are a way of measuring angles using the radius of the circle. Unlike degrees, which divide a whole circle into 360 parts, radians are directly linked to the circle's properties.
A radian essentially measures the radius relative to the circle's circumference:

• One radian is the angle created when the arc length is exactly equal to the radius.
• There are \( 2\pi \) radians in a complete circle, as the circumference of a circle is \( 2\pi r \).

This measure is beneficial in mathematics because it simplifies many formulas and equations, such as those involving trigonometry and calculus. In circle geometry, radians provide a more natural way of understanding angles concerning arc lengths, making computations like the one in your exercise straightforward by using the relationship between linear and radial measurements.

Angle Conversion

Converting angles from degrees to radians is a crucial step in many calculations involving circle geometry. Sometimes we are given angles in degrees and need them in radians to use in formulas like the one used to find arc lengths.
The conversion process is straightforward and uses the relationship:

• \( 180^{\circ} = \pi \text{ radians} \)
• So, to convert degrees to radians, multiply by \( \frac{\pi}{180} \).

For example, in your exercise, the task was to find the arc's length for an \( 80^{\circ} \) angle. To do so, the angle needed conversion to radians. By multiplying \( 80^{\circ} \) by \( \frac{\pi}{180} \), you obtained \( \frac{4\pi}{9} \) radians. This conversion was essential to use in the arc length formula, giving a result that accurately reflects the measurement within the circle's geometry.