Question

Moving Ships Two ships are steaming away from a point \(O\) along routes that make a \(120^{\circ}\) angle. Ship \(A\) moves at 14 knots (nautical miles per hour; a nautical mile is 2000 yards). Ship \(B\) moves at 21 knots. How fast are the ships moving apart when \(O A=5\) and \(O B=3\) nautical miles? 29.5 knots

Short answer

The ships are moving apart at a speed of 29.5 knots.

Step-by-step solution

Identify given information and draw the diagram

Plot the point O and draw two rays extending from it that create a 120-degree angle. Label the end points of those rays as A and B. Now, we have a triangle OAB, with |OA| = 5, |OB|=3 and angle AOB being 120 degrees. Also note the speeds at which A and B are moving away.

Apply Law of Cosines

To understand how quickly the ships are moving apart, we need to determine the rate of change of the distance between them. This requires an application of the Law of Cosines. Using the Law of Cosines, for a triangle with sides of lengths a, b, and c (opposite to angles A, B, C respectively), we know \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Here in triangle OAB, a = 5, b = 3 and C = 120. Calculate the distance AB.

Differentiate the Cosine Law on both sides

Next step is to differentiate the Cosine Law taking derivative with respect to time, in order to obtain the rate at which the separation distance is increasing. Use chain rule for this differentiation.

Substitute the given values

Once you have differentiated the equation in Step 3, substitute the values of \( \dot{a}\), \( \dot{b}\), a, b, and c (which are the rates of Ships A and B, and lengths OA, OB, and AB respectively) into it.

Solve equation for \( \dot{c}\) i.e., rate of separation

After substituting the values, simplify the equation to solve for \(\dot{c}\), which represents the speed at which the ships are moving apart. This rate will be the solution to the problem