10. Given that P and Q are two points which lie on the parallel of latitudo of \( 60^{\circ} \mathrm{N} \). The longitudes of P and Q are \( 30^{\circ} \mathrm{E} \) and \( 150^{\circ} \mathrm{W} \) respectively. Calculate their shortest distance measured along the surface of the earth.

To find the shortest distance between points P (30°E) and Q (150°W) on the latitude of 60°N, we first need to convert the longitudes into a common format. The longitude of Q can be converted as follows: \( 150^{\circ} \text{W} \) is equivalent to \( -150^{\circ} \). Now we calculate the difference in longitudes: \[ \Delta \text{Longitude} = 30^{\circ} - (-150^{\circ}) = 30^{\circ} + 150^{\circ} = 180^{\circ} \] Since they both lie on the same parallel (60°N), we can use the cosine of the latitude to find the actual distance on the surface of the earth: \[ \text{Distance} = \Delta \text{Longitude} \times \frac{\pi}{180} \times R \times \cos(\text{Latitude}) \] Here, \( R \) (the radius of the Earth) is approximately \( 6371 \text{ km} \): \[ \text{Distance} = 180^{\circ} \times \frac{\pi}{180} \times 6371 \times \cos(60^{\circ}) \] Since \( \cos(60^{\circ}) = 0.5 \): \[ \text{Distance} = 6371 \times 0.5 = 3185.5 \text{ km} \] Thus, the shortest distance between points P and Q measured along the surface of the Earth is approximately \( 3185.5 \) kilometers.