Question

A ferry transports tourists between three islands. It sails from the first island to the second island, 4.76 km away, in a direction 37.08 north of east. It then sails from the second island to the third island in a direction 69.08 west of north. Finally it returns to the first island, sailing in a direction 28.08 east of south. Calculate the distance between (a) the second and third islands and (b) the first and third islands.

Answer 1

You may simplify the solution if draw a figure representing all the transitions:

Calculate the distance between the islands. Just break the vectors into x- and y- components:

$$A_x=A\text{ cos}\alpha=3.798\text{ km},\\ A_y=A\text{ sin}\alpha=2.870 \text{ km}.\\ B_x=B\text{ sin}\beta=0.9341B,\\ B_y=B\text{ cos}\beta=0.3571B.\\ C_x=C\text{ sin}\gamma=0.4707C,\\ C_y=C\text{ cos}\gamma=0.8823C.\\$$

Also, from the diagram, we see that the algebraic sum of all these vectors is zero:

$$A_x-B_x+C_x=0,\\ A_y+B_y-C_y=0.$$

Now substitute all components with their right parts above:

$$x|\space3.798-0.9341B+0.4707C=0,\\ y|\space2.870+0.3571B-0.8823C=0.$$

The solution of this system (and distances between is

$$B=7.167\text{ km},\\ C=6.154\text{ km}.$$