Question

The latitudes of Philadelphia and Ankara, Turkey, are the same \(\left(40^{\circ}\right)\), with the first at longitude \(75^{\circ} \mathrm{W}\) and the second at longitude \(33^{\circ} \mathrm{E}\). Calculate the distance between Philadelphia and Ankara along the latitude circle, by first calculating the radius of that circle, using 25,000 miles for the circumference of the earth. Then calculate the distance along a great circle, by noting that the chord connecting the two cities can be thought of as a chord of that circle as well as a chord of the latitude circle. (Hint: You will have to convert the chords to the appropriate sines to make this calculation.)

Short answer

Answer: The approximate distance between Philadelphia and Ankara along the latitude circle is 5209.54 miles, while the distance along the great circle is approximately 8050.44 miles.

Step-by-step solution

Calculate the radius of the earth

The earth's circumference is given as 25,000 miles. To find the radius, we can use the circumference formula. Circumference = \(2 \pi r\) Where, r is the radius and \(\pi \approx 3.1416\). Rearranging the formula to solve for the radius, we get: r = \(\frac{Circumference}{2 \pi}\) Plug in the given value of the earth's circumference and solve for the radius: r = \(\frac{25000}{(2)(3.1416)}\) r ≈ 3978.72 miles

Calculate the radius of the latitude circle

Since we already have the radius of the earth, we can find the radius of the latitude circle. We can use the following formula, where \(R_L\) is the radius of the latitude circle, and \(R_E\) is the radius of the earth: \(R_L = R_E \times \cos{(Latitude)}\) As both cities share the same latitude (\(40^{\circ}\)), we can plug in this value and the calculated earth's radius: \(R_L = 3978.72 \times \cos{(40^{\circ})}\) \(R_L ≈ 3040.95\) miles

Find the angle subtended by the arc connecting the two cities

Since Philadelphia is at longitude \(75^{\circ} \mathrm{W}\) and Ankara is at longitude \(33^{\circ} \mathrm{E}\), the angle subtended by the arc between them is the difference between these two longitudes: Angle = \(75^{\circ} + 33^{\circ} = 108^{\circ}\).

Calculate the distance along the latitude circle

Now that we have the radius of the latitude circle and the angle, we can calculate the distance along the latitude circle using the following arc length formula: Arc length = \(R_L \times angle\) (in radians) First, convert degrees to radians: \(108^{\circ} \times \frac{\pi}{180^{\circ}} = \frac{27 \pi}{45}\) radians Now, calculate the arc length (distance along the latitude circle): Arc length = \((3040.95) *\frac{27 \pi}{45} ≈ 5209.54\) miles

Calculate the distance along a great circle

We can use the sine theorem for spherical trigonometry to find the chord length of the great circle: \(\sin{\frac{c}{2}} = \sin{\frac{a}{2}}\sin{\frac{b}{2}}\) Where \(c\) is the angle subtended by the arc connecting the cities along the great circle, \(a\) is the angle subtended by the arc connecting the cities along the latitude circle, and \(b\) is the co-latitude angle. Since Philadelphia and Ankara have the same latitude of \(40^{\circ}\), their co-latitude angle \(b\) is \(90^{\circ} - 40^{\circ} = 50^{\circ}\). Rearranging the sine theorem for \(c\), we get: \(c = 2 \sin^{-1}{(\sin{\frac{a}{2}}\sin{\frac{b}{2}})}\) Plug in the values we found: \(c = 2 \sin^{-1}{(\sin{\frac{108^{\circ}}{2}}\sin{\frac{50^{\circ}}{2}})} = 116.05^{\circ}\) (approximately) Now, we can use the arc length formula to find the distance along a great circle: Distance = \(R_E \times angle\) (in radians) Convert degrees to radians: \(116.05^{\circ} \times \frac{\pi}{180^{\circ}} \approx 2.02392\) radians Calculate the distance along a great circle: Distance = \((3978.72) * (2.02392) ≈ 8050.44\) miles So, the distance between Philadelphia and Ankara along the latitude circle is approximately 5209.54 miles, while the distance along a great circle is approximately 8050.44 miles.

Key concepts

Great Circle Distance

When traveling on the surface of a sphere, such as Earth, the shortest route between two points is along a great circle. A great circle is formed by the intersection of the sphere with a plane that passes through its center. In the context of Earth, great circle routes are often used in navigation and aviation because they represent the minimum distance between two geographically distant points.

For instance, in the given exercise, we aimed to find the great circle distance between Philadelphia and Ankara, which are on the same latitude but different longitudes. The solution involved spherical trigonometry and the use of the sine theorem to calculate the shortest distance along the Earth's surface. By converting the angle subtended by this arc to radians and multiplying it by the Earth's radius, we obtained an estimation of the great circle distance. Such calculations are crucial for plotting efficient travel routes and understanding Earth's geography.

Latitude Circle Radius Calculation

The Earth is not a perfect sphere but rather an oblate spheroid. This shape causes the radius of a latitude circle—an imaginary line that wraps around the Earth parallel to the equator—to vary depending on the latitude. The exercise we're referring to illustrates how to calculate this variable radius, which is essential when determining distances along a line of constant latitude, or when mapping and surveying geographical regions.

The formula provided in the step by step solution involves taking the cosine of the latitude angle and multiplying it by the Earth's radius to get the radius of the latitude circle at that specific latitude. This relationship accounts for the Earth's curvature and ensures more accurate measurements. Specifically, as we approach the poles, latitude circles become smaller, a fact that must be accounted for in precise calculations involving Earth's geometry.

Arc Length Computation

Arc length computation is the process of determining the distance along a curved path, which is particularly relevant in the context of circles and spheres. To compute the arc length along a latitude circle, as demonstrated in the exercise, it's necessary to know the radius of the circle and the angle that the arc subtends at the center of the Earth, in radians.

The step by step solution to the exercise involved calculating the arc length by multiplying the radius of the latitude circle by the angle in radians. In scenarios that necessitate arc length computation, such as determining the distance between cities at the same latitude but different longitudes, this mathematical operation is crucial. By mastering arc length computation, learners can quantify distances on spherical surfaces like the Earth, aiding in a multitude of real-world applications from geography to engineering.