Question

Yellowknife, Northwest Territories, and Crowsnest Pass, Alberta, lie along the \(114^{\circ} \mathrm{W}\) line of longitude. The latitude of Yellowknife is \(62.45^{\circ} \mathrm{N}\) and the latitude of Crowsnest Pass is \(49.63^{\circ} \mathrm{N}\). Consider Earth to be a sphere with radius \(6400 \mathrm{km}\). a) Sketch the information given above using a circle. Label the centre of Earth, its radius to the equator, and the locations of Yellowknife and Crowsnest Pass. b) Determine the distance between Yellowknife and Crowsnest Pass. Give your answer to the nearest hundredth of a kilometre. c) Choose a town or city either where you live or nearby. Determine the latitude and longitude of this location. Find another town or city with the same longitude. What is the distance between the two places?

Short answer

The distance between Yellowknife and Crowsnest Pass is approximately 1432.32 km. Choose a nearby city's latitude and longitude, and find another place with the same longitude to calculate the distance.

Step-by-step solution

Sketch the Circle

Draw a circle representing Earth. Mark the center of the circle as the center of Earth. Draw the equator as a horizontal line through the center. Mark Yellowknife at latitude \(62.45^{\circ} \mathrm{N}\) and Crowsnest Pass at latitude \(49.63^{\circ} \mathrm{N}\), both on the \(114^{\circ} \mathrm{W}\) longitude.

Find the Latitude Difference

Calculate the difference in latitude between the two locations: \(62.45^{\circ} \mathrm{N} - 49.63^{\circ} \mathrm{N} = 12.82^{\circ}\).

Convert Latitude Difference to Radians

Convert the latitude difference from degrees to radians. Use the conversion factor \(\pi/180\): \[ 12.82^{\circ} \times \frac{\pi}{180} = 0.2238 \text{ radians} \].

Calculate the Distance

Using the arc length formula, calculate the distance between the two locations: \[ d = r \times \theta \], where \[ d \text{ is the distance, } r \text{ is the radius of the Earth (6400 km), and } \theta \text{ is the radian measure of the latitude difference} \]. \[ d = 6400 \text{ km} \times 0.2238 = 1432.32 \text{ km} \].

Information for Part c

Choose a nearby town or city to find its latitude and longitude. Select another place with the same longitude and calculate the distance between them using the same method as above.

Key concepts

latitude and longitude

Latitude and longitude are coordinate systems used to locate places on Earth's surface. Latitude measures the distance north or south of the equator and is expressed in degrees.

Longitude measures the distance east or west of the prime meridian, also in degrees.

For example, Yellowknife is at latitude \(62.45^{\circ} \, \text{N}\) and longitude \(114^{\circ} \, \text{W}\). Crowsnest Pass is at latitude \(49.63^{\circ} \, \text{N}\) on the same longitude of \(114^{\circ} \, \text{W}\). Knowing these angles helps us find specific points on Earth.
The equator is at \(0^{\circ}\) latitude, while the prime meridian is at \(0^{\circ}\) longitude.

sphere geometry

To solve distance problems on Earth, we often treat it as a sphere. A sphere is a 3D shape where all points on the surface are equidistant from the center. This is critical when calculating distances between two points on Earth.

In our example exercise, Earth's radius is considered to be 6400 km.
When drawing a circle to represent Earth, label the center and draw its radius. Yellowknife and Crowsnest Pass should be marked on the circle at their respective latitudes along the same longitude.

• The center of the circle represents the Earth's center.
• The distance from the center to the equator is the Earth’s radius.
• The locations of Yellowknife and Crowsnest Pass can be plotted as points on the circle at specific angles from the equator.

arc length formula

The arc length formula finds the distance between two points on a circle. The formula is \(d = r \, \theta\), where:

• \(d\) is the distance.
• \(r\) is the radius of the circle (in our problem, Earth’s radius, 6400 km).
• \(\theta\) is the central angle in radians between the two points.

In our exercise, we've calculated the latitude difference between Yellowknife and Crowsnest Pass. This difference gives us the central angle. By converting this angle to radians, we apply it to the arc length formula to find the distance.

radian measure conversion

To use the arc length formula effectively, angles must be in radians. Degrees need to be converted to radians using the conversion factor \(\frac{\pi}{180}\). This is crucial because direct multiplication with Earth's radius requires radian measurement.

• Calculate the difference in degrees: \(12.82^{\circ}\)
• Apply the conversion: \12.82^{\circ} \times \frac{\pi}{180} = 0.2238 \, \text{radians}\

With \(0.2238\) radians as \(\theta\), plug this into the arc length formula: \d = 6400 \, \text{km} \times 0.2238 = 1432.32 \, \text{km}\
This results in the distance between Yellowknife and Crowsnest Pass.