Question

There is a pole of height 40 m at the top of a mountain. At a point on the ground level, the angles of elevation of the top and base of the pole are 60° and 45° respectively. Find the height of the mountain.

Answer 1

Given:

Pole height = 40 m on top of the mountain.

From a point on ground, angle of elevation to base of pole mountain top = 45° and to top of pole = 60°.

Step-wise calculation:

1. Let h = height of the mountain and x = horizontal distance from the observation point to the mountain.

From tan 45° = 1 = `h/x`, so x = h.

2. From tan 60° = `sqrt(3)`

= `"Height of top above ground"/x`

= `(h + 40)/x`

Substitute x = h:

`sqrt(3) = (h + 40)/h`

= `1 + 40/h`

3. Solve for h:

`sqrt(3) - 1 = 40/h`

⇒ `h = 40/(sqrt(3) − 1)`

Rationalize:

`h = (40(sqrt(3) + 1))/((sqrt(3) - 1)(sqrt(3) + 1))`

= `(40(sqrt(3) + 1))/2`

= `20(sqrt(3) + 1)`

4. Numerical value:

`sqrt(3) ≈ 1.732`

⇒ h ≈ 20 × 2.732 = 54.64 m to two decimal places.

The height of the mountain is h = `20(sqrt(3) + 1)` m ≈ 54.64 m.