Problem 37 - Algebra and Trigonometry with Analytic Geometry

Question

Measuring distances on Earth The distance between two points \(A\) and \(B\) on Earth is measured along a circle having center \(C\) at the center of Earth and radius equal to the distance from \(C\) to the surface (see the figure). If the diameter of Earth is approximately 8000 miles, approximate the distance between \(A\) and \(B\) if angle \(A C B\) has the indicated measure: (a) \(60^{\circ}\) (b) \(45^{\circ}\) (c) \(30^{\circ}\) (d) \(10^{\circ}\) (e) \(1^{0}\)

Step-by-Step Solution

Answer

(a) 4188.79 miles, (b) 3141.59 miles, (c) 2094.40 miles, (d) 698.13 miles, (e) 69.81 miles.

Step 1: Understand the Earth's Radius

The problem provides the diameter of Earth as 8000 miles. First, find the radius, since the distance along the surface depends on the circumference, which in turn relies on the radius.\[\text{Radius, } r = \frac{\text{Diameter}}{2} = \frac{8000}{2} = 4000 \text{ miles}\]

Step 2: Understand Arc Length Formula

The distance between two points along the surface of a circle is the length of the arc. For a circle, the arc length \(s\) can be calculated as:\[s = r \cdot \theta\]where \(r\) is the radius and \(\theta\) is the angle in radians. Convert the given angles in degrees to radians.

Step 3: Convert Angles from Degrees to Radians

To calculate the arc length, convert the angle from degrees to radians. Use the conversion:\[1^\circ = \frac{\pi}{180} \text{ radians}\]For each angle given, multiply by \(\frac{\pi}{180}\) to get it in radians.

Step 4: Calculate Arc Length for 60 Degrees

Convert \(60^\circ\) to radians:\[60^\circ = 60 \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}\]Calculate the arc length:\[s = 4000 \times \frac{\pi}{3} \approx 4188.79 \text{ miles}\]

Step 5: Calculate Arc Length for 45 Degrees

Convert \(45^\circ\) to radians:\[45^\circ = 45 \times \frac{\pi}{180} = \frac{\pi}{4} \text{ radians}\]Calculate the arc length:\[s = 4000 \times \frac{\pi}{4} \approx 3141.59 \text{ miles}\]

Step 6: Calculate Arc Length for 30 Degrees

Convert \(30^\circ\) to radians:\[30^\circ = 30 \times \frac{\pi}{180} = \frac{\pi}{6} \text{ radians}\]Calculate the arc length:\[s = 4000 \times \frac{\pi}{6} \approx 2094.40 \text{ miles}\]

Step 7: Calculate Arc Length for 10 Degrees

Convert \(10^\circ\) to radians:\[10^\circ = 10 \times \frac{\pi}{180} = \frac{\pi}{18} \text{ radians}\]Calculate the arc length:\[s = 4000 \times \frac{\pi}{18} \approx 698.13 \text{ miles}\]

Step 8: Calculate Arc Length for 1 Degree

Convert \(1^\circ\) to radians:\[1^\circ = 1 \times \frac{\pi}{180} = \frac{\pi}{180} \text{ radians}\]Calculate the arc length:\[s = 4000 \times \frac{\pi}{180} \approx 69.81 \text{ miles}\]

Key Concepts

Earth radius

The radius of Earth is vital in calculating distances like the one between two points along its surface. The radius is half of the diameter, representing the distance from Earth's center to its surface. We know Earth's diameter is approximately 8000 miles. Therefore, to find the radius, we divide the diameter by two:

\[\text{Radius, } r = \frac{8000}{2} = 4000 \text{ miles}\]

This value is crucial as it helps determine how far points are from one another over the surface and aids in calculations involving the circle geometry, like finding the arc length.

degree to radian conversion

In circle-related calculations, it's often necessary to work in radians rather than degrees. This is because many formulas, like the arc length formula, function naturally with radians.

To convert degrees into radians, use the following simple conversion formula:

• 1 degree = \(\frac{\pi}{180}\) radians

For example, converting 60 degrees to radians involves multiplying by this factor:

\[60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \text{ radians}\]

This conversion is essential in performing accurate calculations when dealing with circles on a global scale, such as those seen in geophysics.

circumference of Earth

The circumference of Earth is another crucial concept in understanding how distances are calculated along its surface. The circumference is the total distance around a circle, calculated using the formula:

\[C = 2\pi r\]

Given Earth's radius of 4000 miles, calculate the circumference as follows:

\[C = 2 \pi \times 4000 = 8000\pi \text{ miles}\]

This complete circle's length helps us understand arc segments' distances, essential for measuring distances between locations on Earth.

circle geometry

Circle geometry includes several key concepts that enable us to calculate arc lengths, among other things. One prime concept is the idea that an angle at the circle's center, measured in radians, relates directly to the arc length.

• The arc length \(s\) is determined by \(s = r \cdot \theta\)
• \(r\) is the circle's radius
• \(\theta\) is the angle in radians

Using this relationship, once an angle is known in radians, compute the arc of a circle segment conveniently. For instance, with a 60-degree angle, we find the arc length by:

\[s = 4000 \cdot \frac{\pi}{3} \approx 4188.79 \text{ miles}\]

In this manner, circle geometry principles are applied practically to compute distances along Earth's surface.