The bearing of the lighthouse is N 68 degress E from a ship 43 miles from the lighthouse. How far north of the ship is the lighthouse?
draw a sketch
cos 68 = x/43
To determine how far north of the ship the lighthouse is, we need to find the north component of the distance. Given that the bearing of the lighthouse is N 68 degrees E from the ship, we can conclude that the bearing of the lighthouse from due north is 90° - 68° = 22°.
Using trigonometry, we can calculate the north component as follows:
North Component = Distance * sin(Bearing from North)
Plugging in the values:
North Component = 43 miles * sin(22°)
Calculating:
North Component ≈ 43 * 0.3746 ≈ 16.1 miles
Therefore, the lighthouse is approximately 16.1 miles north of the ship.
To calculate how far north of the ship the lighthouse is, we need to break down the bearing given in the question. The bearing is N 68 degrees E, which means the angle starts from the north (N) and rotates clockwise towards the east (E).
First, let's draw a diagram to visualize the situation. Let's assume that the ship is at point A, and the lighthouse is at point B, which is 43 miles away. We want to find the distance from point A to the northern direction of point B.
N
^
|
|
| B (lighthouse)
|
|______________________> E
A (ship)
Now, we can use trigonometry to find the distance north of the ship. We can split the distance of 43 miles into two components: the northward component and the eastward component.
Using the angle provided (68 degrees), we can find the northward component by calculating the adjacent side of a right triangle. The adjacent side represents the distance east of the ship.
cos(68) = adjacent / hypotenuse
cos(68) = x / 43 miles
Solving for x (adjacent):
x = cos(68) * 43 miles
Now, to find the distance north of the ship, we need to find the opposite side of the triangle. The opposite side represents the distance north of the ship.
Using the pythagorean theorem, we can find the opposite side:
opposite^2 + adjacent^2 = hypotenuse^2
opposite^2 + (cos(68) * 43)^2 = 43^2
Solving for opposite:
opposite = sqrt(43^2 - (cos(68) * 43)^2)
Now, let's calculate the distance north of the ship:
opposite = sqrt(43^2 - (cos(68) * 43)^2)
opposite ≈ 22.782 miles
Therefore, the lighthouse is approximately 22.782 miles north of the ship.