A ship sails 30 kilometers 70 degrees north of east. It then turns 110 degrees to the west, giving it a new heading of 0 degrees north of west.

After sailing another 30 kilometers, how far is the boat (in kilometers) from where it started?

how can going 70 N of E, turning 110 deg to west give a new heading 0 deg n of W?

No

To solve this problem, let's break down the steps taken by the ship:

Step 1: The ship sails 30 kilometers 70 degrees north of east.
Step 2: The ship turns 110 degrees to the west.
Step 3: The ship sails another 30 kilometers.

To calculate the distance of the ship from where it started, we can use vector addition.

Step 1: The ship sails 30 kilometers 70 degrees north of east.
To find the horizontal and vertical components of this step:
Horizontal component = 30 km * cosine(70°)
Vertical component = 30 km * sine(70°)

Horizontal component = 30 km * cos(70°)
Horizontal component ≈ 30 km * 0.3420
Horizontal component ≈ 10.26 km

Vertical component = 30 km * sin(70°)
Vertical component ≈ 30 km * 0.9397
Vertical component ≈ 28.19 km

So, after Step 1, the ship has moved approximately 10.26 km horizontally and 28.19 km vertically.

Step 2: The ship turns 110 degrees to the west.
Since the ship is turning towards the west, this step does not change the position relative to the starting point. Therefore, the distance from the starting point remains the same.

Step 3: The ship sails another 30 kilometers.
Following the same logic as in Step 1, we can find the horizontal and vertical components of this step.

Horizontal component = 30 km * cosine(0°)
Horizontal component = 30 km * cos(0°)
Horizontal component = 30 km * 1
Horizontal component = 30 km

Vertical component = 30 km * sine(0°)
Vertical component = 30 km * sin(0°)
Vertical component = 30 km * 0
Vertical component = 0 km

So, after Step 3, the ship has moved 30 km horizontally and 0 km vertically.

Now, we can add up the horizontal and vertical components to find the total displacement from the starting point.

Total horizontal displacement = Horizontal component from Step 1 + Horizontal component from Step 3
Total horizontal displacement = 10.26 km + 30 km
Total horizontal displacement ≈ 40.26 km

Total vertical displacement = Vertical component from Step 1 + Vertical component from Step 3
Total vertical displacement = 28.19 km + 0 km
Total vertical displacement = 28.19 km

Using the Pythagorean theorem, we can find the total displacement.

Total displacement = √[(Total horizontal displacement)^2 + (Total vertical displacement)^2]
Total displacement = √[(40.26 km)^2 + (28.19 km)^2]
Total displacement ≈ √[1620.27 km^2 + 794.16 km^2]
Total displacement ≈ √[2414.43 km^2]
Total displacement ≈ 49.14 km

Therefore, the ship is approximately 49.14 kilometers from where it started.

To find the distance of the boat from where it started, we can break down the movements into components and use vector addition.

First, let's represent the initial movement of the ship in terms of north and east components. The ship sails 30 kilometers at an angle of 70 degrees north of east. We can find the north and east components using trigonometry.

North component = 30 km * cos(70 degrees)
East component = 30 km * sin(70 degrees)

Next, we need to represent the second movement of the ship after it turns 110 degrees to the west in terms of north and west components. Since the new heading is 0 degrees north of west, we can consider it as moving directly west.

West component = 30 km

Now, let's find the total north and west components of the ship's movements.

Total north component = North component - West component
Total west component = East component

Finally, we can find the distance from where the ship started using the Pythagorean theorem.

Distance = sqrt((Total north component)^2 + (Total west component)^2)

By plugging in the values, we can calculate the distance.