Two ships leave San Francisco at the same time. One travles 40 degrees W of N at the speed of 20 knots. The other travels 10 degrees W of south at a speed of 15 knots. How far apart are they after 11 hours?

The first one will have travelled 220 nautical miles, the second one 165 n miles.

the angle between their paths is 130º

so dist^2 = 220^2 + 165^2 - 2(220)(165)cos130
= ..
= ..

I am sure you can do the arithmetic.

To determine the distance between the two ships after 11 hours, we need to calculate the distance travelled by each ship.

Ship 1, which is traveling at 40 degrees west of north at a speed of 20 knots, forms a right triangle with the north direction. We can use trigonometry to find the distance traveled by this ship.

The distance traveled by Ship 1 (d1) is given by the formula:
d1 = speed × time

Substituting the given values into the equation:
d1 = 20 knots × 11 hours = 220 nautical miles

Now let's calculate the distance traveled by Ship 2.

Ship 2 is traveling at 10 degrees west of south at a speed of 15 knots. To determine the distance traveled by this ship, we can once again form a right triangle. The angle between the ship's direction and the north direction is 90 degrees - 10 degrees = 80 degrees.

The distance traveled by Ship 2 (d2) is given by the following formula:
d2 = speed × time

Substituting the given values into the equation:
d2 = 15 knots × 11 hours = 165 nautical miles

Now we have the distances traveled by both ships. To find the distance between them, we can use the Pythagorean theorem because the two distances form the two sides of a right triangle.

The distance between the two ships is given by the formula:
distance = √(d1^2 + d2^2)

Substituting the calculated values into the equation:
distance = √(220^2 + 165^2)
distance = √(48400 + 27225)
distance = √75625
distance ≈ 275.12 nautical miles

Therefore, after 11 hours, the two ships are approximately 275.12 nautical miles apart.

To find the distance between two ships after 11 hours, we need to calculate the positions of both ships after 11 hours and then find the distance between these positions.

Let's start with the first ship. It travels 40 degrees west of north at a speed of 20 knots. This means it is moving towards the northwest direction. We can represent the ship's velocity vector as v1 = 20 knots at an angle of 40 degrees west of north. To find the displacement vector (change in position) of the ship after 11 hours, we can use the formula:

Displacement = Velocity × Time

The magnitude of the displacement vector is given by:

|Displacement1| = v1 × 11

To find the second ship's displacement vector, we need to consider that it travels 10 degrees west of south at a speed of 15 knots. This means it is moving towards the southwest direction. Representing its velocity vector as v2 = 15 knots at an angle of 10 degrees west of south, we can use the same displacement formula:

|Displacement2| = v2 × 11

After calculating both |Displacement1| and |Displacement2|, we can determine the distance between the two positions using the Pythagorean theorem since they form a right-angled triangle.

Distance = √(|Displacement1|^2 + |Displacement2|^2)

Plugging in the values, we can find the distance between the two ships after 11 hours.