Question

A boat is headed at 115° and is traveling downstream at 28 mph. The stream is flowing S38°E at 11 mph. In what direction is the boat traveling? At what rate can it travel in still water?

Answer 1

A boat is headed at $115{}^{\circ}$ and is traveling downstream at 28 mph. The stream is flowing S38°E at 11 mph. In what direction is the boat traveling? At what rate can it travel in still water?

Solution

The direction is the boat traveling is at $115{}^{\circ}$ to the direction of the stream (S38°E). We have

$$\mathrm{S}38{}^{\circ}\mathrm{E} + 115{}^{\circ} = \mathrm{W}13{}^{\circ}\mathrm{S}.$$

The velocity of a boat $(\overrightarrow{V_1})$ is a vector sum of vectors of velocity in still water $(\overrightarrow{V_0})$ and the stream $(\vec{V})$:

$$\overrightarrow{V_1} = \overrightarrow{V_0} + \vec{V}.$$

According to the cosine theorem

$$V_0^2 = V_1^2 + V^2 - 2VV_1 \cos 115 = V_1^2 + V^2 + 2VV_1 \cos 65.$$

So

$$V_0 = \sqrt{28^2 + 11^2 + 2 * 11 * 28 * \cos 65} = 34.14 \text{ mph}.$$

Answer: W13°S; 34.14 mph.