Towns A and B are on the equator. A is on longitude 30°W. The time at B is 17:05 when it is 14:05 at A. Find longitude X. A ship sailed east from A to B at 450 knots. Find the travel time.

a) Find the longitude X.

Step 1: Calculate the time difference between town B and town A. Time at B = 17:05 hours Time at A = 14:05 hours Time difference = $17:05 - 14:05 = 3$ hours.

Step 2: Determine the direction of town B relative to town A. Since the time at B is later than the time at A, town B is to the east of town A.

Step 3: Convert the time difference into a longitude difference. The Earth rotates $360^\circ$ in 24 hours. Therefore, $1$ hour corresponds to $\frac{360^\circ}{24} = 15^\circ$ of longitude. Longitude difference = $3 hours \times 15^\circ/hour = 45^\circ$.

Step 4: Calculate the longitude of town B (X). Town A is at $30^\circ$W. Town B is $45^\circ$ to the east of A. Starting from $30^\circ$W, moving $30^\circ$ east brings us to the Prime Meridian ($0^\circ$). We need to move an additional $45^\circ - 30^\circ = 15^\circ$ further east. This means town B is at $15^\circ$E. So, X = $15^\circ$E.

The longitude X is $\boxed{15^\circE}$.

b) Find the time it took the ship to travel from A to B.

Step 1: Calculate the total angular distance between town A and town B. Town A is at $30^\circ$W. Town B is at $15^\circ$E. The angular distance is the sum of their longitudes because they are on opposite sides of the Prime Meridian. Angular distance = $30^\circ + 15^\circ = 45^\circ$.

Step 2: Convert the angular distance into nautical miles. On the equator, $1^\circ$ of longitude corresponds to $60$ nautical miles. Distance = $45^\circ \times 60 nautical miles/degree = 2700 \text{ nautical miles}$.

Step 3: Calculate the time taken using the formula Time = Distance / Speed. Given speed = $450$ knots ($450$ nautical miles/hour). Time taken = $\frac{2700 nautical miles}{450 nautical miles/hour}$ Time taken = $6$ hours.

The time it took the ship to travel from A to B is $\boxed{6 hours}$.