Problem 75 - Algebra and Trigonometry

Question

Nautical Miles Find the distance along an arc on the sur- face of the earth that subtends a central angle of 1 minute \(\left(1 \text { minute }=\frac{1}{60} \text { degree). This distance is called a nautical mile. }\right.\) (The radius of the earth is 3960 \(\mathrm{mi}\) .)

Step-by-Step Solution

Answer

Approximately 1.1508 miles.

Step 1: Understanding the problem

We need to find the distance along an arc on the surface of the Earth that subtends a central angle of 1 minute. A nautical mile is defined as this distance. Also, we know the Earth's radius is given as 3960 miles.

Step 2: Convert angle from minutes to degrees

First, convert the 1 minute angle into degrees. Since 1 degree equals 60 minutes, \(1\) minute is equal to \(\frac{1}{60}\) degree.

Step 3: Use the formula for arc length

The formula for the arc length \(L\) along a circle is given by: \[ L = \theta \times r \] where \(\theta\) is in radians and \(r\) is the radius. Convert \(\theta = \frac{1}{60}\) degrees to radians using the formula \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Step 4: Convert degrees to radians

Convert \(\frac{1}{60}\) degrees to radians: \[ \frac{1}{60} \times \frac{\pi}{180} = \frac{\pi}{10800} \text{ radians} \]

Step 5: Calculate Arc length

Substitute \( r = 3960 \) miles and \( \theta = \frac{\pi}{10800} \) radians into the arc length formula: \[ L = 3960 \times \frac{\pi}{10800} \] Calculate the value to find \( L \).

Step 6: Compute final value

Performing the multiplication: \( L \approx 3960 \times 0.000290888... \approx 1.1508 \) miles. Thus, the distance is approximately 1.1508 miles.

Key Concepts

Arc Length

The concept of arc length is essential for measuring distances along the curved surface of a circle or, in this case, the Earth. An arc is just a piece of a circle's circumference. To find its length, we use the formula:
\[ L = \theta \times r \]where:

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

When you have the central angle in radians, you multiply it by the radius to get the arc length. In practical applications like nautical miles, this helps in measuring distances on the Earth, which is essentially a gigantic circle. The arc length gives us a straightforward way to understand and calculate these curved distances.

Central Angle

A central angle is an angle whose vertex is at the center of a circle, and whose sides extend out to the circle's edge, forming an arc. Central angles play a crucial role in calculating arc lengths. They are usually measured in degrees or radians.
In the exercise, you have a small central angle of 1 minute. One minute is \( \frac{1}{60} \) of a degree, demonstrating how even tiny angles span significant distances across large circles like Earth. This specific angle measurement is key in defining a nautical mile, as a one-minute central angle on Earth corresponds to a nautical mile along its surface. Working with such precise angles is vital in navigation to ensure accurate travel distances between locations on the globe.

Conversion to Radians

Radians are a unit of angular measurement based on the radius of a circle. In many mathematical applications, using radians simplifies calculations, especially those involving trigonometric functions.
To convert from degrees to radians, we use the formula:\[\text{radians} = \text{degrees} \times \frac{\pi}{180}\]For the central angle in our exercise, which is \(\frac{1}{60}\) degrees, the conversion is: \[\frac{1}{60} \times \frac{\pi}{180} = \frac{\pi}{10800} \text{ radians}\]This conversion is essential because the formula for arc length requires the angle to be in radians. The conversion from degrees to radians allows us to appropriately apply the formula, ensuring an accurate calculation of the nautical mile.