A lighthouse is 14 miles east of a dock. A ship sails N33degreesE from the dock. What will be its bearing from the lighthouse after having sailed 10 miles?
After the 10 mile sailing, let the distance between boat and lighthouse be x
by the cosine law, you can find x
x^2 = 14^2+10^2 - 2(14)(10)cos57° , (90°-33°=57°)
Once you have x, you can find a second angle using the sine law, and voila!, easy after that.
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To find the ship's bearing from the lighthouse after sailing 10 miles, we can use trigonometry and vector addition.
First, let's visualize the situation. Draw a diagram with the dock, lighthouse, and the ship's initial and final positions.
Since the ship sails N33°E from the dock, we can decompose this direction into north and east components. N33°E means the ship is moving 33° east of the north direction.
Using trigonometry, we can find the north and east components of the ship's movement. The north component can be found by multiplying the distance traveled (10 miles) by the sine of the angle (33°). The east component can be found by multiplying the distance traveled (10 miles) by the cosine of the angle (33°).
North component = 10 miles × sin(33°)
East component = 10 miles × cos(33°)
Next, we need to find the coordinates of the initial position of the ship. Since the lighthouse is 14 miles east of the dock, and the ship sails N33°E, the ship's position relative to the dock can be represented by the vector [14, 0].
To find the final position of the ship, we add the north and east components of the movement to the initial position vector.
Final position = Initial position + [North component, East component]
Finally, to find the bearing from the lighthouse, we calculate the angle between the direction of the ship relative to the dock and the direction of the lighthouse relative to the dock.
Bearing = arctan(East component / North component)
Plug in the values for the north and east components, and calculate the bearing.
Once you perform these calculations, you will find the bearing of the ship from the lighthouse after having sailed 10 miles.