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?
D = 5mi/h * 1.5h = 7.5mi = Dist. north.
Dist. = 7.5mi / Cos(35) = 9.2mi =
Dist. from tower.
To find the distance between the ship and the tower at 1:30 pm, we need to use some trigonometry. Here's how we can solve the problem:
Step 1: Determine the time elapsed from noon to 1:30 pm.
Given that the ship noticed the tower at noon and the bearing at 1:30 pm, the time elapsed is 1 hour and 30 minutes.
Step 2: Convert the time elapsed from minutes to hours.
Since we'll be using the speed of the ship in mph, we need to convert the time to hours. 30 minutes is half of an hour, so we add 0.5 to the total hours elapsed.
Step 3: Calculate the distance traveled by the ship.
To find the distance traveled by the ship, we multiply its speed (5.0 mph) by the elapsed time in hours.
Distance = Speed × Time
Distance = 5.0 mph × (1.0 hour + 0.5 hour) = 5.0 mph × 1.5 hours = 7.5 miles
Step 4: Determine the relative bearing of the tower.
The bearing of the tower is given as S 35° E. The "S" indicates a south direction, and the "E" indicates an east direction. Therefore, the tower is located 35° to the east from the south direction.
Step 5: Use trigonometry to calculate the distance between the tower and the ship.
We can use trigonometry to find the distance between the tower and the ship. We have a right triangle where the angle between the ship's heading and the bearing to the tower is 35°, and the distance traveled by the ship is the base of the triangle.
We can use the tangent function to calculate the distance.
Distance = Tan(angle) × Base
Distance = Tan(35°) × 7.5 miles
Using a calculator, the tangent of 35° is approximately 0.7002.
Therefore, the distance between the ship and the tower at 1:30 pm is approximately 0.7002 × 7.5 miles = 5.25 miles.
So, the ship is approximately 5.25 miles away from the tower at 1:30 pm.