Solution:

In the figure

AB is the tower

BD and BC are the shadow of the tower in two situations

Consider BD = x m and AB = h m

In triangle ABD

tan 45⁰ = h/x

So we get

1 = h/x

h = x ….. (1)

In triangle ABC

tan 30⁰ = h/(x + 10)

So we get

1/√3 = h/(x + 10)

Using equation (1)

h√3 = h + 10

h (√3 – 1) = 10

We know that

h = 10/(√3 – 1)

It can be written as

h = [10 (√3 + 1)]/ [(√3 – 1) (√3 + 1)]

By further calculation

h = (10√3 + 1)/ 2

So we get

h = 5 (1.73 + 1)

h = 5 × 2.73

h = 13.65 m

Therefore, the height of the tower is 13.65 m.