From the top of a 7 m high building, the angle of elevation of the top of a tower is 60°, and the angle of depression of its foot is 45°. Find the height of the tower.

Let AB be the building and CE be the tower

Given the height of the building, AB = 7m

To find: Height of tower, i.e. CE

From the top of building, angle of elevation of the top of tower = 60°

Hence, ∠EAD = 60°

The angle of depression of the foot of the tower = 45°

Hence, ∠CAD = 45°

Since AB and CD are parallel

CD = AB = 7m

Also,

AD and BC are parallel

So, AD = BC

Since tower and building are vertical to the ground

∠ABC = 90° and ∠EDA = 90°

Now, AD and BC are parallel

Taking AC as transversal

∠ACB = ∠DAC = 45° [Alternate angles]

Now, In Δ ABC, we have

⇒ BC = 7m …(i)

Since BC = AD

So, AD = 7m

Now, In ΔADE, we have

[from(i)]

⇒ ED = 7√3 m

Hence, the height of the tower, CE = ED + DC

= 7√3 + 7

= 7(√3 +1)m