Geometry

Trigonometry

Medium

From the top of a building 78 m high, the angles of depression of the bottom and the top of a tower are observed to be 45° and 30° respectively. Determine the approximate height of the tower so observed?

30 m

33 m

35 m

38 m

✅ Correct Option: 2

Let the building height be 78 m, the tower height be $h$ m, and the horizontal distance between them be $d$ m.

The angle of depression to the bottom of the tower is 45°.

The angle of depression to the top of the tower is 30°.

For the angle of depression to the bottom of the tower:

$\tan(45°) = \frac{78}{d}$

$1 = \frac{78}{d}$

$d = 78$ m

For the angle of depression to the top of the tower, the vertical distance from the top of the building to the top of the tower is $(78 - h)$ m.

$\tan(30°) = \frac{78 - h}{d}$

$\frac{1}{\sqrt{3}} = \frac{78 - h}{78}$

$78 - h = \frac{78}{\sqrt{3}}$

$78 - h = \frac{78}{1.732}$

$78 - h = 45.03$

$h = 78 - 45.03$

$h = 33$ m

Therefore, the height of the tower is approximately 33 m.

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