The law of sines and the trigonometric ratios for sine indicates that we get;
B ≈ 33.42°
C ≈ 69.58°
c ≈ 37.51
12. 5.72 mi
What is the law of sines; The law of sines, also known as the sine rule states that the ratio of the sine of an angle in a triangle to the length of the side facing the angle is the same for the three angles in the triangle
sin(A)/a = sin(B)/b = sin(C)/c
The values of the angles indicates that we get;
A = 77°, a = 39, and b = 22
The law of sines indicates that we get;
(sin(77°)/39 = (sin(B))/22
sin(B) = 22 × (sin(77 Degrees)/39
22 × (sin(77 Degrees)/39 ≈ 0.55
sin(B) ≈ 0.55
B ≈ arcsin(0.55)
arcsin(0.55) ≈ 33.42
B ≈ 33.42°
C = 180 - 77 - 33.42
180 - 77 - 33.42 = 69.58°
C = 69.58°
c/sin(69.58°) ≈ 39/sin(77°)
c ≈ sin(69.58°) × 39/sin(77°)
c ≈ 37.51
12. The speed of the boat due east = 9 miles per hour
The distance traveled by the boat in 30 minutes is; 0.5 × 9 mph = 4.5 miles
The interior angles of the triangle formed by the positions of the boat and the light are; (90 - 70)° = 20°, (90 + 63)° = 153°, and (180 - 20 - 153)° = 7°
The length of the path from the initial position to the lighthouse, l, can therefore be found as follows;
l/(sin(153°)) = 4.5/sin(7°)
l = sin(153°) × 4.5/sin(7°)
sin(153°) × 4.5/sin(7°) ≈ 16.73
l ≈ 16.73 miles
The angle formed by the path l and the horizontal, obtained using the 70° formed at the initial position of the boat, the height, d, and the horizontal distance from the lighthouse to a point just below the boat is; 90° - 70° = 20°
The trigonometric ratio for sines indicates that we get;
d = l × sin(20°)
Therefore;
d ≈ 16.73 × sin(20°)
16.73 × sin(20°) ≈ 5.72
d ≈ 5.72 miles