Lamp Post Height Calculation Using Trigonometry

This problem involves calculating the height of a lamp post using the concept of angles of elevation and trigonometry.

Problem Setup

Let the height of the lamp post be '$h$' meters. Let the initial distance of the person from the base of the lamp post be '$x$' meters.

When the person walks 30 meters towards the lamp post, the new distance from the base is '$(x - 30)$' meters.

We have two angles of elevation:

• Initial angle: $30^{\circ}$
• Final angle: $60^{\circ}$

Applying Trigonometric Ratios

We use the tangent ratio ($\tan$) since we have the opposite side (height '$h$') and adjacent sides (distances '$x$' and '$(x-30)$').

Scenario 1: Initial position

The angle of elevation is $30^{\circ}$ and the distance is '$x$'.

$ \tan(30^{\circ}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x} $

We know $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$.

$ \frac{1}{\sqrt{3}} = \frac{h}{x} $

Rearranging this equation gives: $ x = h\sqrt{3} \quad (*)$

Scenario 2: Final position

The angle of elevation is $60^{\circ}$ and the distance is '$(x - 30)$'.

$ \tan(60^{\circ}) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{x-30} $

We know $\tan(60^{\circ}) = \sqrt{3}$.

$ \sqrt{3} = \frac{h}{x-30} $

Rearranging this equation gives: $ h = (x-30)\sqrt{3} \quad (**)$

Solving for the Height

Now, substitute the value of '$x$' from equation ($*$) into equation ($**$):

$ h = (h\sqrt{3} - 30)\sqrt{3} $

Distribute $\sqrt{3}$ on the right side:

$ h = h(\sqrt{3} \times \sqrt{3}) - (30 \times \sqrt{3}) $

$ h = 3h - 30\sqrt{3} $

Rearrange the terms to solve for '$h$':

$ 3h - h = 30\sqrt{3} $

$ 2h = 30\sqrt{3} $

Divide by 2:

$ h = \frac{30\sqrt{3}}{2} $

$ h = 15\sqrt{3} $

The height of the lamp post is $15\sqrt{3}$ meters.