Problem 87 - Precalculus

Question

Find the exact length of the are made by the indicated central angle and radius of each circle. $$\theta=8^{\circ}, r=1500 \mathrm{km}$$

Step-by-Step Solution

Answer

The exact length of the arc is \(66.6667\pi \text{ km}.\)

Step 1: Convert Degrees to Radians

The angle \(\theta = 8^{\circ}\) must be converted into radians since arc length is calculated using radians. Use the conversion \(1^{\circ} = \frac{\pi}{180} \text{ radians}\). Thus, \(\theta = 8^{\circ} = 8 \times \frac{\pi}{180} = \frac{2\pi}{45} \text{ radians}\).

Step 2: Use the Arc Length Formula

The formula for the arc length \(s\) is \(s = r\theta\), where \(r\) is the radius and \(\theta\) is the angle in radians. Substitute the given radius and the converted angle into the formula: \(s = 1500 \, km \, imes \, rac{2\pi}{45}.\)

Step 3: Calculate Arc Length

Simplify the expression: \(s = \frac{1500 \, km \, imes \ 2\pi}{45}.\) Divide 1500 by 45 to get \ 33.3333. The exact length of the arc is \(s = 66.6667\pi\, km\).

Key Concepts

Radians Conversion

When dealing with angular measurements, it's crucial to understand the difference between degrees and radians. A circle has 360 degrees, but mathematically, it's more convenient to work with radians. One radian is the angle created when the radius of a circle is wrapped around its circumference. The conversion between degrees and radians is essential when you want to calculate arc length for a given angle. To convert an angle from degrees to radians, you use the formula \[1^{\circ} = \frac{\pi}{180} \text{ radians.}\]This formula tells us how many radians are in one degree. To convert our given angle of 8 degrees, we multiply:\[8^{\circ} \times \frac{\pi}{180} = \frac{2\pi}{45} \text{ radians.}\]This conversion allows us to use the arc length formula since the calculation requires the angle to be in radians.

Central Angle

A central angle is an angle whose vertex is at the center of a circle, and its sides are radii that extend to the circumference. This angle plays a critical role in calculating arc lengths and understanding the geometry of circles. When we talk about a central angle in radians, we're referring to how much 'turn' it represents on the circle.

• Understanding the central angle: In a complete circle, the central angle in radians would be \(2\pi\), equivalent to 360 degrees.
• This means that smaller central angles represent smaller portions of the circle, which directly affects the length of the arc.

For the exercise, the central angle given was 8 degrees, which we've converted into \[\frac{2\pi}{45} \text{ radians.}\]This conversion simplifies the calculation process for determining how much of the circle is being utilized by the given angle.

Arc Length Formula

The arc length is a portion of the circle's circumference associated with a specific central angle. This length is an important measurement in many fields such as navigation and construction. To calculate the arc length, we use a simple but powerful formula:\[s = r \theta\]Where:

• \(s\) is the arc length,
• \(r\) is the radius of the circle,
• \(\theta\) is the central angle in radians.

For our problem, with a radius of 1500 km and a central angle of \[\frac{2\pi}{45} \text{ radians,}\]the formula becomes:\[s = 1500 \times \frac{2\pi}{45}.\]Simplifying the multiplication, we calculate the length of the arc to find it's \[66.6667\pi \text{ km.}\]This showcases the direct relationship between the radius, central angle, and the arc length, allowing us to solve real-world problems involving circular shapes.