Problem 51 - Algebra 2

Question

Geography Assume that Earth is a sphere with radius 3960 miles. A town is at latitude \(32^{\circ} \mathrm{N} .\) Find the distance in miles from the town to the North Pole. (Hint: Latitude is measured north and south from the equator.)

Step-by-Step Solution

Answer

The surface distance from the town to the North Pole on Earth's surface is approximately 4007.12 miles.

Step 1: Understanding Latitudes

Latitudes are measured from the equator, which is at 0°. The North Pole is at 90° latitude, and the South Pole is at -90° or 270°. Since the town is located at a latitude of \(32^{\circ} \mathrm{N} \), its angular location north from the equator is 32 degrees.

Step 2: Find Angular Distance from North Pole

The latitude of the North Pole is 90°, so subtract the latitude of the town from 90 to find the angular distance from the town to the North Pole. \(90^{\circ} - 32^{\circ} = 58^{\circ} \)

Step 3: Convert Angular Distance to Rotation

To find the surface distance from the town to the North Pole, convert the angular distance to radians. There are \( \frac{180}{\pi} \) degrees in one radian, so multiply the angular distance by \( \frac{\pi}{180} \) to convert to radians. \(58^{\circ} \times \frac{\pi}{180} = 1.012 radians\)

Step 4: Find Surface Distance

On a sphere, the surface distance between two points equals the radius of the sphere times the angle between those two points, measured in radians. Therefore, multiply the radius of the Earth by the angular distance to get the surface distance from the town to the North Pole. Resulting in \(3960 miles \times 1.012 radians = 4007.12 miles\)

Key Concepts

Latitude Calculation

Latitude refers to the angle measured north or south from the Earth's equatorial plane. Understanding latitude is crucial in calculating distances on a spherical surface like Earth. The equator is at 0 degrees latitude, where the Earth is widest. The North Pole is at 90 degrees north of the equator and the South Pole is at 90 degrees south, though geographically it could be marked as -90 degrees for calculation purposes.

To determine the latitude of a point such as our town, we note that the location is specified as 32° N (north), indicating a point 32 degrees from the equator in the northern hemisphere. This measurement helps us to indicate how far the town is from the poles, providing a foundation for further distance calculations.

Angular Distance

Once we know the latitude of a location, we can calculate the angular distance between it and another point on Earth. The angular distance is simply the difference in latitude between two points. In our case, we are interested in the distance from the town at 32° N to the North Pole at 90° N.

To find the angular distance, subtract the town's latitude from that of the North Pole:

• North Pole at 90°
• Town at 32° N
• Angular distance = 90° - 32° = 58°

This 58° is important because it tells us how far in angular terms the town is from the North Pole.

Radians Conversion

Degrees are commonly used for measuring angles, but radians are often more useful for mathematical calculations involving circular geometry. A full circle is 360 degrees, which is equivalent to 2π radians. Thus, any angular measurement in degrees can be converted to radians using the relationship:

• Convert degrees to radians:
• Radians = Degrees × (π / 180)

For our specific problem, we convert 58° to radians:

• 58° × (π / 180) ≈ 1.012 radians

This conversion helps set the stage for determining the arc length or surface distance along the sphere corresponding to this angular change.

Surface Distance on a Sphere

To calculate the surface distance on a sphere between two points, we use the concept of arc length in a circle. The arc length of a circle, or sphere in this case, is given by the product of the radius of the circle and the angle in radians subtended by the arc at the circle's center:

• ext{Arc Length (Surface Distance)} = ext{Radius} × ext{Angle in Radians}

Given the Earth's radius is 3960 miles, and using our computed angle in radians (1.012 radians), we find:

• Surface Distance = 3960 miles × 1.012 radians ≈ 4007.12 miles

Therefore, the surface distance from the town to the North Pole along the Earth's surface is approximately 4007.12 miles. This method accurately reflects how one travels over the curved surface of the Earth rather than a straight line through its interior.