A fishing boat leaves a marina and follows a course of S65°W at 8 knots for 15min. Then the boat changes to a new course of S36°W at 5 knots for 2hr.

Question

a) How far is the boat from the marina?

b) What bearing/course should the boat follow to return straight to the marina?

Answer 1

You will need to solve for all the sides and angles for 3 right triangles in this problem. First, let's figure out how far the boat went in each direction (which will be the hypotenuse for triangles 1 and 2, respectively).

Going S65°W at 8 knots for 15 minutes (0.25 hrs): 8 knots (1.852 km/hr / 1 knot) = 14.816 km/hr 14.816 km/hr (0.25 hrs) = 3.704 km

Going S36°W at 5 knots for 2 hours): 5 knots (1.852 km/hr / 1 knot) = 9.26 km/hr 9.26 km/hr (2 hrs) = 18.52 km

The first right triangle will be the boat going S65°W. For this triangle, your initial angle is 65°. Since this is a right triangle, we also know the second angle will be 90°, and all three angles will sum to 180° for any triangle.

triangle 1: 180° - 90° - 65° = 25°

The second right triangle will be the boat going S36°W.

triangle 1: 180° - 90° - 36° = 54°

Now we can solve for the 2 other sides for the triangles using the law of sines, a / (sin (A)) = b / (sin (B)) = c / (sin (C)), and the pythagorean theorem (a² + b² = c²).

triangle 1: a = (3.704) • ((sin(65)) / (sin (90)) = 3.357 3.357² + b² = 3.704² b = 1.565

triangle 2: a = (18.52) • ((sin(36)) / (sin (90)) = 10.89 10.89² + b² = 18.52² b = 14.98

If we lined up the two triangles correctly, we can get the final location of the boat. Using the final location of the boat, along with the initial location, we can set up a third right triangle. By adding the bases and heights of the other two triangles, we can determine the base and height of the new triangle.

bases added: 3.357 + 10.89 = 14.25

heights added: 1.565 + 14.98 = 16.55

Using the pythagorean theorem, we can determine the hypotenuse for this triangle, which is equivalent to the distance the boat is from the marina

14.25² + 16.55² = c² c = 21.84, the boat is 21.84 km from the marina

Lastly, using the law of sines, we can determine the bearing.

16.55 / sin(ß) = 21.84 / sin (90°) sin(ß) = 0.75779 sin^-1(0.75779) = 49.27°

Going straight north, to get to the marina would require going east to some degree. The amount east can be determined by subtracting that value from 90°, which gives the bearing: N40.73°E