Find the distance between New York \( \mathrm{C} 42^{\circ} \mathrm{N} \) \( 74^{\circ} \) W) and Rome \( \left(42^{\circ} \mathrm{N}, 12^{\circ} \mathrm{E}\right) \) measured alo a parallel of lafitude.

To find the distance between New York and Rome measured along the parallel of latitude \(42^{\circ}N\), we can use the formula for calculating distances along a circle of latitude. The formula is: \[ D = R \cdot \cos(\phi) \cdot \Delta \lambda \] where \(D\) is the distance, \(R\) is the Earth's radius (approximately 6371 km), \(\phi\) is the latitude (42 degrees), and \(\Delta \lambda\) is the difference in longitude (from \(-74^{\circ}\) to \(12^{\circ}\)). First, convert the longitudes into the same degree format: \[ \Delta \lambda = 12 - (-74) = 12 + 74 = 86^{\circ} \] Next, convert the latitude from degrees to radians since the cosine function requires radians: \[ \phi \, \text{(in radians)} = 42^{\circ} \times \frac{\pi}{180} \approx 0.733 \, \text{radians} \] Calculate the cosine of the latitude: \[ \cos(42^{\circ}) \approx \cos(0.733) \approx 0.743 \] Now, substitute everything into the distance formula: \[ D \approx 6371 \cdot 0.743 \cdot \left(\frac{86 \times \pi}{180}\right) \] Calculating \(\Delta \lambda\) in radians: \[ \Delta \lambda \, \text{(in radians)} = 86^{\circ} \times \frac{\pi}{180} \approx 1.500 \] Thus, calculating \(D\): \[ D \approx 6371 \cdot 0.743 \cdot 1.500 \approx 7117 \, \text{km} \] So, the distance between New York and Rome along the parallel of latitude \(42^{\circ}N\) is approximately 7117 kilometers!