Question

A vertical tower stands on a horizontal plane and is surmounted by a vertical flagstaff of height 5 m. At a point on the plane, the angles of elevation of the bottom and the top of the flagstaff are respectively 30° and 60°. Find the height of the tower.

Answer 1

Given:

Let the height of the tower = H (m).

Height of the flagstaff = 5 m.

From a point on the ground, angle of elevation to bottom of flagstaff (top of tower) = 30° and to top of flagstaff = 60°.

Step-wise calculation:

1. Let the horizontal distance from the observation point to the base of the tower = d.

2. From the bottom of the flagstaff (top of tower):

`tan 30° = H/d`

⇒ `1/sqrt(3) =H/d`

⇒ `d = Hsqrt(3)`

3. From the top of the flagstaff:

`tan 60^circ = (H + 5)/d`

⇒ `sqrt(3) = (H + 5)/d`

⇒ `d = (H + 5)/sqrt(3)`

4. Equate the two expressions for d:

`Hsqrt(3) = (H + 5)/sqrt(3)`

5. Multiply both sides by `sqrt(3)`:

3H = H + 5

6. Solve:

2H = 5

⇒ `H = 5/2` 

= 2.5 m

The height of the tower is 2.5 m.