A boat sails 150 km on a bearing of 042° and then 250 km due east. How far is the boat from the starting point?
We can solve this problem using the Law of Cosines.
First, we need to draw a diagram to visualize the problem:
```
042°
|
|
| 150 km
|
|
|
----------------------
250 km to the east
```
We can see that we have a triangle with two sides and the included angle given. We need to find the third side, which is the distance between the starting point and the ending point of the boat's journey.
Let's call this third side "d".
Using the Law of Cosines, we have:
d^2 = 150^2 + 250^2 - 2(150)(250)cos(138°)
(Note that we found the angle opposite the side we want using 180° - 42° - 90° = 48°, and then used the fact that the sum of angles in a triangle is 180° to find the remaining angle to be 138°.)
Simplifying this equation, we get:
d^2 = 90000 + 62500 - 75000(-0.6691)
d^2 = 150625 + 50218.92
d^2 = 200843.92
Taking the square root of both sides, we get:
d = 448.1 km (rounded to one decimal place)
Therefore, the boat is approximately 448.1 km away from the starting point.
We can solve this problem by breaking it down into two components: the northward distance traveled by the boat and the eastward distance traveled by the boat.
First, let's calculate the northward distance traveled. The boat sailed 150 km on a bearing of 042°, which means it traveled in a north-east direction. To calculate the northward distance, we need to find the component of the total distance traveled that is in the northward direction.
Using trigonometry, we can calculate the northward distance traveled by multiplying the total distance traveled (150 km) by the cosine of the angle of the bearing. The cosine of 42° is approximately 0.7431.
Northward distance = Total distance * cosine(of bearing angle)
Northward distance = 150 km * 0.7431
Northward distance ≈ 111.465 km
Next, let's calculate the eastward distance traveled. The boat traveled 250 km due east, so the entire distance is in the eastward direction.
Therefore, the total distance from the starting point is the sum of the northward distance and the eastward distance.
Total distance = Northward distance + Eastward distance
Total distance = 111.465 km + 250 km
Total distance ≈ 361.465 km
Hence, the boat is approximately 361.465 km from the starting point.