Geometry

Trigonometry

Easy

From a point exactly midway between the foot of two towers P and Q, the angle of elevation of their tops are 30° and 60°, respectively. The ratio of the heights of tower P to that of Q is:

1:3

1:2

$1:2\sqrt{3}$

$1:\sqrt{3}$

✅ Correct Option: 1

Two towers P and Q have a point O exactly midway between their bases. From point O, the angle of elevation to the top of tower P is 30° and to the top of tower Q is 60°.

Let $d$ = distance from point O to the foot of each tower

Let $h_1$ = height of tower P

Let $h_2$ = height of tower Q

For tower P, using the tangent ratio:

$\tan(30°) = \frac{h_1}{d}$

Since $\tan(30°) = \frac{1}{\sqrt{3}}$ :

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

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

For tower Q, using the tangent ratio:

$\tan(60°) = \frac{h_2}{d}$

Since $\tan(60°) = \sqrt{3}$ :

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

$h_2 = d\sqrt{3}$

The ratio of heights is:

$h_1 : h_2 = \frac{d}{\sqrt{3}} : d\sqrt{3}$

Dividing both sides by $d$ :

$= \frac{1}{\sqrt{3}} : \sqrt{3}$

Multiplying both sides by $\sqrt{3}$ :

$= 1 : 3$

Therefore, the ratio of the heights of tower P to tower Q is $1:3$ .

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Solution

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