Question

A 200-foot tall monument is located in the distance. From a window in a building, a person determines that the angle of elevation to the top of the monument is 13°, and that the angle of depression to the bottom of the tower is 4°. How far is the person from the monument? (Round your answer to three decimal places.)

Answer 1

Answer:

665 ft

Explanation:

Let d be the distance from the person to the monument. Note that d is perpendicular to the monument and would make 2 triangles with the monuments, 1 up and 1 down.

The side length of the up right-triangle knowing the other side is d and the angle of elevation is 13 degrees is

$dtan13^0 = 0.231d$

Similarly, the side length of the down right-triangle knowing the other side is d and the angle of depression is 4 degrees

$dtan4^0 = 0.07d$

Since the 2 sides length above make up the 200 foot monument, their total length is

0.231d + 0.07d = 200

0.301 d = 200

d = 200 / 0.301 = 665 ft

Answer 2

The distance of the person from the monument is 184.608 ft.

Considering the triangle in the image provided, the distance of the person from the monument is calculated as follows;

$tan(\theta ) = \frac{b}{y} \\\\y = \frac{b}{tan(\theta)} \\\\y = \frac{b}{tan(13)} \ --(1)$

From the bigger triangle;

$tan(13) = \frac{200- b}{y} \\\\y = \frac{200- b}{tan(13)} \ --(2)$

solve equation (1) and (2) together;

$\frac{b}{tan(4)} = \frac{200-b}{tan(13)} \\\\tan(4)(200 - b) = tan(13)(b)\\\\13.985 - 0.07b = 0.23b\\\\0.3b = 13.985\\\\b = \frac{13.985}{0.3} \\\\b = 46.62 \ ft$

The distance of the person from the monument is calculated as;

$y = \frac{b}{tan(13)} \\\\y = \frac{46.62}{tan(13)} \\\\y = 184.608 \ ft$

Thus, the distance of the person from the monument is 184.608 ft.

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