A ship sets sail from port A (86 ^o N, 56 ^o W) for port B (86 ^o N, 64 ^o W), which is close by. Find the…

Analysis

To find the distance between two points on the same parallel of latitude, we use the formula for the arc length of a small circle. Since both ports are at the same latitude ($86^\circ$N), the ship is traveling along a circle of latitude.

Concept

The distance ($d$) between two points on the same latitude is given by:

Where:

• $\theta$ = The difference in longitude between the two points.
• $R$ = Radius of the Earth (6370 km).
• $\alpha$ = The latitude of the points ($86^\circ$).
• $\pi$ = 3.142.

Step-by-Step Solution

Step 1: Find the difference in longitude ($\theta$) The ports are at $56^\circ$W and $64^\circ$W. Since they are both in the same hemisphere (West), we subtract them:

Step 2: Apply the formula Substitute the values into the distance formula:

Step 3: Calculate the values

• $\cos(86^\circ) \approx 0.069756$
• $d = \frac{8}{360} \times 2 \times 3.142 \times 6370 \times 0.069756$
• $d = 0.02222 \times 6.284 \times 6370 \times 0.069756$
• $d \approx 0.1396 \times 6370 \times 0.069756$
• $d \approx 889.25 \times 0.069756$
• $d \approx 62.02 \text{ km}$

Rounding to the nearest whole number, we get 62 km.

Conclusion

The distance covered by the ship is approximately 62 km. This matches option A.

Exam Tip

When calculating distances on the Earth's surface, always check if the points are on the same latitude or the same longitude. If they are on the same latitude, remember to include the $\cos(\text{latitude})$ factor in your calculation, as the radius of the circle of latitude is smaller than the Earth's radius ($R \cos \alpha$).