Question

A survey team is trying to estimate the height of a mountain above a level plain. From one point on the plain, they observe that the angle of elevation to the top of the mountain is 30 ∘ . From a point 2000 feet closer to the mountain along the plain, they find that the angle of elevation is 33 ∘ . How high (in feet) is the mountain?

Answer 1

Answer:

10406.5937 ft

Step-by-step explanation:

In the figure attached the graph of the problem is shown.

Data for triangle ABC:

∠A = 30°

∠B = 180° - 33° = 147°,

segment AB = 2000 feet long.

∠C = 180° - 30° - 147° = 3°

From law of sines:

AB/sin(C) =  AC/sin(B)

2000/sin(3) =  AC/sin(147)

AC = [2000/sin(3)]*sin(147)

AC = 20813.1875 ft

Using now triangle ADC:

sin(A) = CD/AC

CD = sin(A)*AC

CD = sin(30)*20813.1875

CD = 10406.5937 ft

Answer 2

The angle of elevation is simply the angle from the level plain, and the line of sight.

The height of the mountain is 10406.58 feet.

I've attached a diagram that illustrates the scenario.

From the diagram, we have:

$\angle A = 30$

$AB = 2000$

$\angle ABC = 180 - 33$

$\angle ABC = 147$

$\angle BCA = 180 - \angle A - \angle ABC$

$\angle BCA = 180 - 30 - 147$

$\angle BCA = 3$

First, we calculate side length BC as follows:

$\frac{BC}{\sin A} = \frac{AB}{\sin C}$

So, we have:

$BC =\sin A \times \frac{AB}{\sin C}$

This gives:

$BC =\sin (30) \times \frac{2000}{\sin (3)}$

$BC =19107.3$

Side length CD (i.e. the height of the mountain) is calculated as follows:

$\sin (\angle DBC) = \frac{CD}{BC}$

$\sin (33) = \frac{CD}{19107.3}$

Make CD the subject

$CD = \sin (33) \times 19107.3$

$CD = 10406.58$

Hence, the height of the mountain is 10406.58 feet.

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