Two ships are steaming straight away from a point O along routes that make a 150-degree angle. Ship A moves at 5 knots (nautical miles per hour). Ship B moves at 14 knots. How fast are the ships moving apart when ship A is 12 nautical miles away from point O and ship B is 15 nautical miles away from point O?

Ships are moving apart at_______knots

We can solve this problem using the Law of Cosines and related rates. Here's a step-by-step breakdown:

### Step 1: Define Variables

- Let \( A \) be the distance of Ship A from point \( O \).
- Let \( B \) be the distance of Ship B from point \( O \).
- Let \( \theta = 150^\circ \) be the angle between the two ships' routes.
- Let \( d \) be the distance between the two ships.
- We need to find \( \frac{dd}{dt} \), the rate at which the distance between the ships is changing.

### Step 2: Use the Law of Cosines

The Law of Cosines gives the relationship between the distances:

\[
d^2 = A^2 + B^2 - 2AB \cdot \cos(\theta)
\]

Substitute the known values for \( A \), \( B \), and \( \theta \):

\[
d^2 = 12^2 + 15^2 - 2 \cdot 12 \cdot 15 \cdot \cos(150^\circ)
\]

We know that \( \cos(150^\circ) = -\frac{\sqrt{3}}{2} \), so:

\[
d^2 = 144 + 225 + 2 \cdot 12 \cdot 15 \cdot \frac{\sqrt{3}}{2}
\]

Simplifying:

\[
d^2 = 144 + 225 + 180\sqrt{3}
\]

Now, we take the derivative of both sides with respect to time \( t \) to get the rate at which the distance between the ships is changing. Then

Two ships are steaming straight away from a point O along routes that make a 150-degree angle. Ship A moves at 5 knots (nautical miles per hour). Ship B moves at 14 knots. How fast are the ships moving apart when ship A is 12 nautical miles away from point O and ship B is 15 nautical miles away from point O?

Learn how to calculate the rate at which two ships are moving apart using trigonometry and the Law of Cosines. This problem involves applying related rates to determine the speed of separation when ships A and B are at specific distances from a common point. Suitable for advanced high school or early college students.

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