Geometry

Trigonometry

Medium

The angle of elevation of the top of an unfinished tower at a distance of 75m from its base is 30°. How much higher must the tower be raised so that the angle of elevation of its top at the same point may be 60°?

$25\sqrt{3}$ m

$50\sqrt{3}$ m

30 m

60 m

✅ Correct Option: 2

Let the current height of the tower be $h_1$ and the final height be $h_2$ . The distance from the base is 75m.

Using the tangent ratio in the first triangle with angle of elevation 30°:

$\tan(30°) = \frac{h_1}{75}$

$\frac{1}{\sqrt{3}} = \frac{h_1}{75}$

$h_1 = \frac{75}{\sqrt{3}}$

Rationalizing:

$h_1 = \frac{75}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

$h_1 = \frac{75\sqrt{3}}{3}$

$h_1 = 25\sqrt{3}$ m

Using the tangent ratio in the second triangle with angle of elevation 60°:

$\tan(60°) = \frac{h_2}{75}$

$\sqrt{3} = \frac{h_2}{75}$

$h_2 = 75\sqrt{3}$ m

The additional height required is the difference between the final height and current height:

Additional height $= h_2 - h_1$

$= 75\sqrt{3} - 25\sqrt{3}$

$= 50\sqrt{3}$ m

Therefore, the tower must be raised by $50\sqrt{3}$ meters.

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