A 7 m long flagstaff is fixed on the top of a tower. From a point on the ground, the angles of elevation of the top and bottom of the flagstaff are 45⁰ and 36⁰ respectively. Find the height of the tower correct to one place of decimal

Solution:

Consider TR as the tower and PT as the flag on it

PT = 7 m

Take TR = h and AR = x

Angles of elevation from P and T are 45⁰ and 36⁰

In right triangle PAR

tan θ = PR/AR

Substituting the values

tan 45⁰ = (7 + h)/ x

So we get

1 = (7 + h)/ x

x = 7 + h …. (1)

In right triangle TAR

tan θ = TR/AR

Substituting the values

tan 36⁰ = h/x

So we get

0.7265 = h/x

h = x (0.7265) …… (2)

Using both the equations

h = (7 + h) (0.7265)

By further calculation

h = 7 × 0.7265 + 0.7265h

h – 0.7265h = 7 × 0.7265

So we get

0.2735h = 7 × 0.7265

By division

h = (7 × 0.7265)/ 0.2735

We can write it as

h = (7 × 7265)/ 2735

h = 18.59 = 18.6 m

Hence, the height of the tower is 18.6 m.