A surveyor leaves her base camp and drives 43km on a bearing of 032 degrees. She then drives 28km
on a bearing of 154 degrees. How far is she then
from her base camp and what is her bearing from
it?​

1. The distance from the base camp is 37 Km

2. The bearing from the base camp is 72°

Step-by-step explanation:

Please see attached photo for explanation.

1. Determination of the distance from the base camp.

In the attached photo, x is the distance from the base camp. The value of x can be obtained by using the cosine rule formula. This is illustrated below:

Side opposite A = a = 28 Km

Side opposite C = c = 43 Km

Angle B = 58°

Side opposite B = b = x =?

b² = a² + c² – 2ac Cos B

x² = 28² + 43² – 2 × 28 × 43 Cos 58

x² = 784 + 1849 – 2408 × 0.52991

x² = 2633 – 1276

x² = 1357

Take the square root of both side

x = √1357

x ≈ 37 Km

Thus, the distance from the base camp is 37 Km.

2. Determination of the bearing from the base camp

We'll begin by calculating the value of θ in the attached photo. This can be obtained by using sine rule formula as shown below:

Side opposite A = a = 28 Km

Side opposite B = b = x = 37

Angle B = 58°

Angle A = θ =?

a/Sine A = b/SineB

28/Sine θ = 37/Sine 58

Cross multiply

37 × Sine θ = 28 × Sine 58

Divide both side by 37

Sine θ = 28 × Sine 58 /37

Sine θ = 28 × 0.8480 / 37

Sine θ = 0.6417

Take the inverse of sine

θ = Sine¯¹ 0.6417

θ = 40

Finally, we shall determine the bearing from the base camp. This can be obtained as follow:

Bearing from the base camp = 32 + θ

θ = 40

Bearing from the base camp = 32 + 40

Bearing from the base camp = 72°