Problem 39 - Algebra and Trigonometry

Question

Two ships leave a harbor at the same time. One ship travels on a bearing of \(\mathrm{S} 12^{\circ} \mathrm{W}\) at 14 miles per hour. The other ship travels on a bearing of \(\mathrm{N} 75^{\circ} \mathrm{E}\) at 10 miles per hour. How far apart will the ships be after three hours? Round to the nearest tenth of a mile.

Step-by-Step Solution

Answer

The ships will be approximately 43.8 miles apart after 3 hours.

Step 1: Interpret the Bearings

Let's start with the given bearings. A bearing of S \(12^{\circ}\) W means an angle of \(12^{\circ}\) South of West. This translates to an angle of \(180^{\circ} - 12^{\circ} = 168^{\circ}\) measured counterclockwise from East. A bearing of N \(75^{\circ}\) E means an angle of \(75^{\circ}\) North of East. This translates to an angle of \(75^{\circ}\) measured counterclockwise from East.

Step 2: Calculate the Distances

Use the speed and time to calculate how much distance each ship has traveled in the 3 hours. Use the formula Distance = Speed * Time. Ship 1 has traveled 14 mph * 3 hours = 42 miles. Ship 2 has traveled 10 mph * 3 hours = 30 miles.

Step 3: Apply the Law of Cosines

To find the distance between the ships, we can apply the Law of Cosines since we know the lengths of the two sides of the triangle formed by the paths of the ships and also the included angle. The Law of Cosines formula is \(c = \sqrt{(a^2 + b^2 - 2*a*b*cos(C))}\), where a and b are the sides of the known lengths, C is the included angle and c is the missing side. Substituting our known values, we get \(c = \sqrt{(42^2 + 30^2 - 2*42*30*cos(\angle{168^{\circ} - 75^{\circ}}))}\).

Step 4: Calculate the final Distance

Solve the expression to find the value of c after rounding to the nearest tenth of a mile. This gives the final distance between the two ships after 3 hours.