Problem 95 - Algebra and Trigonometry

Question

How do we measure the distance between two points, \(A\) and \(B,\) on Earth? We measure along a circle with a center, \(C,\) at the center of Earth. The radius of the circle is equal to the distance from \(\mathrm{C}\) to the surface. Use the fact that Earth is a sphere of radius equal to approximately 4000 miles to solve Exercises 93-96. If \(\theta=30^{\circ},\) find the distance between \(A\) and \(B\) to the nearest mile.

Step-by-Step Solution

Answer

The distance between points \(A\) and \(B\) is approximately 2094 miles.

Step 1: Convert the Angle from Degrees to Radians

The angle \(\theta\) given is in degrees. To convert it to radians, the conversion factor \(\frac{\pi}{180}\) is used. Therefore, \(\theta\) in radians is \(30^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6}\) radians.

Step 2: Calculate the Arc Length

To find the length of the arc, or in this case the distance between points \(A\) and \(B\), we use the formula: Arc Length = \(r \cdot \theta\). With the radius \(r\) of the Earth being approximately 4000 miles, and \(\theta = \frac{\pi}{6}\) radians, we substitute these into the formula and solve for Arc Length, giving us Arc Length = \(4000 \times \frac{\pi}{6} = \frac{2000\pi}{3}\) miles. This is the exact answer.

Step 3: Round to the Nearest Mile

Since the question calls for the nearest mile, this requires rounding \(\frac{2000\pi}{3}\) miles to the nearest whole number. Using the approximate value of \(\pi \approx 3.1416\), we get the distance to be approximately 2094 miles.

Key Concepts

Understanding Arc Length

To calculate the distance between two points on Earth through the surface, we measure what is known as the arc length. This is the distance along the circle's curve, not in a straight line. For a sphere like Earth, arc length is found on the circle that has the midpoint of Earth as its center, known as the great circle. It's a part of a circumference pertaining to a sector defined by an angle \(\theta\).
The general formula for finding the arc length \(L\) of a circle is:

• \(L = r \cdot \theta\)

where:

• \(r\) is the radius of the circle.
• \(\theta\) is the angle in radians.

For Earth, when calculating distances like this, the radius \(r\) is roughly 4000 miles.
Once \(\theta\) is in radians, you simply multiply by the radius to find the arc length. This makes understanding and computing distances over a sphere straightforward once you grasp these basic elements.

The Importance of Radian Conversion

Radians are essential for calculating dimensions of circles and circular objects, such as Earth. Why use radians? They're simple in mathematical formulas, especially those involving arcs and angles. Our angle \(\theta = 30^{\circ}\) must be converted to radians for our formula to work.
To convert degrees to radians, we use the conversion formula:

• \(\text{Radians} = \text{Degrees} \times \frac{\pi}{180}\)

This approach uses the mathematical constant \(\pi\), making radians closer tied to the circle's nature. For example:

• \(30^{\circ} \times \frac{\pi}{180} = \frac{\pi}{6}\)

This conversion step is fundamental, as it allows us to use formulas dealing with arcs accurately. Thus, always remember, when diving into circle mathematics, prioritizing radian conversion is crucial for clarity and accuracy.

Exploring the Sphere

A sphere is a perfect instance of three-dimensional symmetry, with all points on its surface equidistant from the center. Earth, for instance, is often approximated as a sphere with a radius of about 4000 miles, making spherical trigonometry essential for geographical calculations.
Sphericity implies:

• Each slice through the center forms a circle.
• Distances measured across the surface, called great-circle distances, involve finding arc lengths.

Great circles allow you to determine the shortest distance between two points on a sphere's surface. This principle is applied in various fields, such as aviation, where true distances over Earth need to consider its spherical shape, avoiding lengthy paths. Appreciating a sphere’s properties helps us visualize travel paths and distances, reinforcing the relevance of trigonometric functions and concepts like arc length on the sphere.