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:
[tex]\angle O = 36.283^{\circ}-15.333^{\circ}[/tex]
[tex]\angle O = 20.950^{\circ}[/tex]
[tex]\angle B = 180^{\circ}-90^{\circ}-(90^{\circ}-15.333^{\circ})[/tex]
[tex]\angle B = 15.333^{\circ}[/tex]
[tex]\angle A = 180^{\circ}-\angle O - \angle B[/tex] (1)
[tex]\angle A = 180^{\circ}-20.950^{\circ}-15.333^{\circ}[/tex]
[tex]\angle A = 143.717^{\circ}[/tex]
By the law of Sine we determine the length of the segment OB:
[tex]\frac{AB}{\sin O} = \frac{OB}{\sin A}[/tex] (2)
[tex]OB = \left(\frac{\sin A}{\sin O}\right)\cdot AB[/tex]
If we know that [tex]\angle A = 143.717^{\circ}[/tex], [tex]\angle O = 20.950^{\circ}[/tex] and [tex]AB = 108\,ft[/tex], then the length of the segment OB is:
[tex]OB = \left(\frac{\sin 143.717^{\circ}}{\sin 20.950^{\circ}} \right)\cdot (108\,ft)[/tex]
[tex]OB \approx 178.747\,ft[/tex]
Lastly, we determine the height of the observation deck by the following trigonometric identity:
[tex]d = OB\cdot \sin B[/tex] (3)
If we know that [tex]OB \approx 178.747\,ft[/tex] and [tex]\angle B = 15.333^{\circ}[/tex], then the height of the observation deck is:
[tex]d = (178.747\,ft)\cdot \sin 15.333^{\circ}[/tex]
[tex]d\approx 47.266\,ft[/tex]
The height of the observation deck is approximately 47.266 feet.
