Question

A man 1.75 metres tall, standing at the foot of a tower sees the top of a hill 40 metres away at an elevation of 60°. On climbing to the top of the tower, he sees the top of the hill at an elevation of 50°. Compute the heights of the hill and the tower.

Answer 1

Let EG be the height of the tower and EF be the height of the man standing at the foot of the tower.

Let AD be the height of the hill.

Let ∠AFC be the angle of elevation made by the man’s eye while seeing the hill from the foot of the tower.

Let GH be the position of the man when he reaches the top of the tower to see the hill.

Let ∠AHB be the angle made by the man’s eye while seeing the hill from the top of the tower.

Let DE be the distance between the tower and the hill.

Let AB = y m, GF = x m

The figure for the given situation with dimensions can be made as follows:

In ΔABH:

Now in ΔACF:

∴Height of the hill = (y + x + 1.75 + 1.75) m

≈ (47.672 + 19.862 + 1.75 + 1.75) m

= 71.034 m

Height of the tower = (x + 1.75) m

≈ (19.862 + 1.75) m

= 21.612 m

Thus, the height of the hill and the tower is approximately equal to 71.034 m and 21.612 m.