Question

From an observation deck, Matt spots two deer to the right of the observation deck, standing 108 feet apart in a field below. The angles of depression to each of the deer are 15o20’ and 36o17’. How high is the observation deck he is standing on?

Answer 1

Answer:

The height of the observation deck is approximately 47.266 feet.

Step-by-step explanation:

We include a geometrical representation of the statement in the image attached below. Let be O the location of the observation deck, and A and B the locations of the two deers, which are 108 feet apart of each other. By knowing that sum of internal angles within triangle equals 180º. The angles O, B and A are now determined:

$\angle O = 36.283^{\circ}-15.333^{\circ}$

$\angle O = 20.950^{\circ}$

$\angle B = 180^{\circ}-90^{\circ}-(90^{\circ}-15.333^{\circ})$

$\angle B = 15.333^{\circ}$

$\angle A = 180^{\circ}-\angle O - \angle B$ (1)

$\angle A = 180^{\circ}-20.950^{\circ}-15.333^{\circ}$

$\angle A = 143.717^{\circ}$

By the law of Sine we determine the length of the segment OB:

$\frac{AB}{\sin O} = \frac{OB}{\sin A}$ (2)

$OB = \left(\frac{\sin A}{\sin O}\right)\cdot AB$

If we know that $\angle A = 143.717^{\circ}$, $\angle O = 20.950^{\circ}$ and $AB = 108\,ft$, then the length of the segment OB is:

$OB = \left(\frac{\sin 143.717^{\circ}}{\sin 20.950^{\circ}} \right)\cdot (108\,ft)$

$OB \approx 178.747\,ft$

Lastly, we determine the height of the observation deck by the following trigonometric identity:

$d = OB\cdot \sin B$ (3)

If we know that $OB \approx 178.747\,ft$ and $\angle B = 15.333^{\circ}$, then the height of the observation deck is:

$d = (178.747\,ft)\cdot \sin 15.333^{\circ}$

$d\approx 47.266\,ft$

The height of the observation deck is approximately 47.266 feet.