An airplane is traveling from New York to Paris, a distance of $5.80 \times 10^{3} \mathrm{km} .$ Ignore the curvature of the Earth. (a) If the cruising speed of the airplane is \(350.0 \mathrm{km} / \mathrm{h},\) how much time will it take for the airplane to make the round-trip on a calm day? (b) If a steady wind blows from New York to Paris at \(60.0 \mathrm{km} / \mathrm{h},\) how much time will the round-trip take? (c) How much time will it take if there is a crosswind of \(60.0 \mathrm{km} / \mathrm{h} ?\)

The distance between New York and Paris is given as \(5.80 \times 10^{3} \mathrm{km}\). For the round-trip, the total distance will be \(2 \times (5.80 \times 10^{3}) \mathrm{km}\). Since the cruising speed of the airplane is given as 350 km/h, the time taken can be calculated as: Time taken = Total Distance / Effective Speed Time taken = \(2 \times (5.80 \times 10^{3} \mathrm{km}) / 350.0 \mathrm{km/h}\) Calculating the time taken: Time taken ≈ 33.14 hours

We need to calculate the effective speed of the airplane while traveling with and against the wind. Speed with wind = Airplane Speed + Wind Speed = 350.0 + 60.0 km/h = 410.0 km/h Speed against wind = Airplane Speed - Wind Speed = 350.0 - 60.0 km/h = 290.0 km/h Now we can calculate the time taken for each leg of the journey: Time taken with wind = Distance / Speed with wind = \((5.80 \times 10^{3}) \mathrm{km} / 410.0 \mathrm{km/h}\) Time taken against wind = Distance / Speed against wind = \((5.80 \times 10^{3}) \mathrm{km} / 290.0 \mathrm{km/h}\) Total time taken = Time taken with wind + Time taken against wind Calculating the total time taken: Total time taken ≈ 33.62 hours

Since a crosswind does not affect the speed in the direction of travel, the effective speed of the airplane remains the same as its cruising speed, which is 350 km/h. Hence, the time taken for a round-trip with a crosswind will be the same as the time taken for a round-trip on a calm day. Time taken = 33.14 hours