Problem 87 - Precalculus with Calculus Previews

Question

Sailing at Sea A nautical mile is defined as the arc length subtended on the surface of the Earth by an angle of measure 1 minute. If the diameter of the Earth is 7927 miles, find how many statute (land) miles there are in a nautical mile.

Step-by-Step Solution

Answer

A nautical mile is approximately 1.1508 statute miles.

Step 1: Calculate Earth's Radius

The diameter of the Earth is given as 7927 miles. Since the radius is half the diameter, we calculate the radius using the formula: \(\text{radius} = \frac{7927}{2}\). Thus, the radius of the Earth is 3963.5 miles.

Step 2: Convert Angle to Radians

We know that 1 nautical mile subtends an angle of 1 minute at the center of Earth. First, convert this angle from minutes to degrees: 1 minute corresponds to \(\frac{1}{60}\) degree since there are 60 minutes in a degree. Hence the angle in degrees is \(\frac{1}{60}\) degrees.

Step 3: Convert Degrees to Radians

We convert the angle from degrees to radians using the conversion factor \(\frac{\pi}{180}\): \(\text{angle in radians} = \frac{1}{60} \times \frac{\pi}{180}\). Therefore, the angle in radians is \(\frac{\pi}{10800}\).

Step 4: Calculate the Arc Length in Statute Miles

The arc length \(s\) in statute miles can be calculated using the formula \(s = r\theta\), where \(r\) is the radius in statute miles and \(\theta\) is the angle in radians. Substituting the values, we have \(s = 3963.5 \times \frac{\pi}{10800}\). This equals approximately 1.1508 statute miles.

Key Concepts

Arc Length Calculation

Understanding the concept of arc length is essential when dealing with any circular movement or curvature, such as measuring distances on the surface of the Earth. The arc length represents the distance along the curved line, comparable to a straight line distance in Euclidean geometry but on a curved surface. For circles, the arc length \(s\) is easily determined by the formula \(s = r\theta\), where \(r\) is the radius of the circle, and \(\theta\) is the angle subtended by the arc at the center of the circle, measured in radians.

In the context of nautical miles, the arc length subtended by 1 minute (1/60th of a degree) on the Earth's surface is what defines this unit of measure. Calculating the arc length using the Earth's radius provides the "nautical mile" in terms of the Earth's curvature, distinctly different from the statute mile, which is based on linear distance measurements.

Radians Conversion

Converting angles into radians is essential for calculating arc lengths and other circular measurements effectively. Radians provide a natural measure for angles because they compare arc length directly to the radius of the circle.

To convert from degrees to radians, you use the conversion factor \(\frac{\pi}{180}\). This factor arises from the fact that a full circle corresponds to 360 degrees or \(2\pi\) radians. Therefore, one degree equals \(\frac{\pi}{180}\) radians. For instance, when dealing with nautical miles and the Earth's curvature, you often convert smaller subdivisions of angles, such as minutes, where 1 degree equals 60 minutes, and hence 1 minute equals \(\frac{1}{60}\) degree.

• The formula is: \(\text{angle in radians} = \frac{\text{angle in degrees} \times \pi}{180}\).
• For this example, \(\frac{1}{60}\) degree converts to \(\frac{\pi}{10800}\) radians.

This conversion is crucial for using the formula \(s = r\theta\) accurately in arc length calculations, aligning perfectly with the natural geometry of circles.

Earth's Radius Calculation

When involving measurements around Earth, knowing or calculating its radius is fundamental. The radius is half of the Earth's diameter. Given an estimated diameter of 7927 miles, the Earth's radius becomes \(\frac{7927}{2} = 3963.5\) miles.

This radius is key to converting angular measures to real-world distances, such as when determining the length of an arc that equates to a nautical mile. Nautical miles are inherently linked to the Earth's sphere, as they use the Earth's curvature for their definition as opposed to flat distances used in statute (land) miles.

• Radius is essential in the formula \(s = r\theta\).
• It bridges geometrical definitions with real-world distance applications, particularly overt spaces such as open water, where traditional land measures may fall short.

Understanding Earth's radius allows us to translate angular units into navigational ones with high accuracy, showcasing how geometry and geography fuse together in practical applications like nautical navigation.