Question

A surveyor sights the far bank of a river at an angle of 110° to the near bank. She then moves 75 feet upriver and sights the same point on the far bank of the river at an angle of 150°. What is the shortest distance across the river?

Answer 1

have you covered the Law of Sines yet?

Answer 2

Yes

Answer 3

ok, if you look at the 150 External angle, the Internal angle, to the left-hand-side of it, will be 180-150, thus 30 degrees

so you have a triangle with 2 internal angles, one is 30 degree, and one is 110
internal angles for a triangle ADD UP to 180, so, the angle "above", would be 180-(110+30), so that'll make it 40

40 has an opposite side, or a side that's facing it off, of 75ft
and what you want is the smaller side/line to cross the river, the smaller line/side to cross the river is the side facing off the internal angle of 30 degrees, thus

$$
c = \text{opposite side to the }30^o \text{ angle}\\
\cfrac{75}{sin(40^o)}=\cfrac{c}{sin(30^o)}
$$

Answer 4

then you just solve for "c"

Answer 5

75/sin(40°) = 116.7 I believe
sin (30°) = 0.5

So that leaves
116.7 = c/0.5
So if I multiply by 0.5 on each side.
I end up with c= 58.34

Answer 6

Does that look right?

Answer 7

yes