Question

A plane flies from base camp to lake a, 205 km away in the direction 20.0° north of east. after dropping off supplies it flies to lake b, which is 175 km at 30.0° west of north from lake
a. determine the distance and direction from lake b to the base camp.

Answer 1

Draw a diagram to illustrate the problem as shown in the figure below.

Camp A  is 20° north of east from the camp, therefore
m∠CAB = 80°
where C => base camp.

Let d = distance from lake B to the base camp, at x° west of south.
Apply the Law of Cosines to determine d.
d² = 205² + 175² - 2*205*175*cos80⁰
     = 6.0191 x 10⁴
d = 245.338 km

Apply the Law of Sines to obtain
$\frac{sin(x+30)}{205}= \frac{sin80^{o}}{245.338}\\ \\ sin(x+30)=( \frac{205}{245.338})sin80^{o}= 0.8229$
x+30 = sin⁻¹ 0.8229 = 55.4°
x = 25.4°

Answer:
The distance from lake B to base camp is 243.3km (nearest tenth).
The direction is 25.4° west of south.