Two ships leave port at the same time and travel straight line distances, the first at 30 km/h and the second at 10 km/h. Two hours later they are 50 km apart. What is the angle between their courses?

So you have three legs of a triangle. Perfect for the law of cosines. make a sketch, then use the law of cosines.

My triangle has sides 60 , 20 and 50 km

and I am finding the angle opposite the side of 50
by cosine law:

50^2 = 20^2 + 60^2 - 2(20)(60)cosØ
cosØ = .625
Ø = appr 51.3°

one goes 60 km

one goes 20 km
the third side is 50 km
divide them all by ten (similar triangle
then law of cosines
5^2 = 6^2 + 2^2 - 2*6*2 * cos A

25 = 36 + 4 - 24 cos A

-15 = -24 cos A
cos A = .625
A = 51.3

To find the angle between the courses of the two ships, we can use the Law of Cosines. But before we do that, let's try to understand the situation and see if we can find any helpful information.

The first ship is traveling at 30 km/h, and the second ship is traveling at 10 km/h. Two hours later, they are 50 km apart. We can assume that both ships are traveling in a straight line.

Let's visualize the situation:

Ship A (30 km/h)
-------------------> (Course of Ship A)

Ship B (10 km/h)
-------------------> (Course of Ship B)

After two hours, the first ship (A) has traveled a distance of 30 km/h * 2 h = 60 km.
The second ship (B) has traveled a distance of 10 km/h * 2 h = 20 km.

Now, the distance between the ships is 50 km. We can imagine this as a triangle, with one ship at each corner and the distance between them as the side connecting them.

Applying the Law of Cosines, we have:

c^2 = a^2 + b^2 - 2ab * cos(C)

Where:
- c is the distance between the ships (50 km)
- a is the distance traveled by ship A (60 km)
- b is the distance traveled by ship B (20 km)
- C is the angle between the courses of the ships (which we want to find)

Plugging in the values, we get:

50^2 = 60^2 + 20^2 - 2 * 60 * 20 * cos(C)

Simplifying the equation:

2500 = 3600 + 400 - 2400 * cos(C)
2400 * cos(C) = 3600 + 400 - 2500
2400 * cos(C) = 1500
cos(C) = 1500 / 2400
cos(C) = 5/8

To find the angle C, we can use the inverse cosine function (cos^(-1)):

C ≈ cos^(-1)(5/8)
C ≈ 39.2°

Therefore, the angle between the courses of the two ships is approximately 39.2°.