A passenger on a ship sailing north at 5.0 mph noticed that at noon a radio tower on land was due east of the slip. At 1:30pm, the bearing of the tower from the ship was S 35deg E. How far was the ship from the tower at 1:30pm?

To find the distance between the ship and the tower at 1:30 pm, we need to use trigonometry and the information given in the problem.

First, let's find the angle between the ship's heading and the line connecting the ship to the tower. At noon, the tower was due east, which means it was at 90 degrees from the ship's heading (north). At 1:30 pm, the bearing of the tower from the ship was S 35 degrees E. This means that the angle between the ship's heading and the line connecting the ship to the tower is 180 degrees - 35 degrees = 145 degrees.

Next, we can calculate the distance between the ship and the tower using trigonometry. We have the angle between the ship's heading and the line connecting the ship to the tower (145 degrees) and the speed of the ship (5.0 mph).

Using the relationship: distance = speed x time, we can calculate the distance the ship traveled between noon and 1:30 pm. In 1.5 hours (the time between noon and 1:30 pm), the ship traveled a distance of 5.0 mph x 1.5 hours = 7.5 miles.

To find the distance between the ship and the tower at 1:30 pm, we can use the cosine rule:

distance^2 = (ship distance to tower)^2 + (ship distance traveled)^2 - 2 x (ship distance to tower) x (ship distance traveled) x cos(angle between the ship's heading and the line connecting the ship to the tower)

Substituting the known values:

(distance)^2 = (ship distance to tower)^2 + (7.5 miles)^2 - 2 x (ship distance to tower) x (7.5 miles) x cos(145 degrees)

Now, we can solve this equation to find the distance between the ship and the tower at 1:30 pm.