Question

If one looks from a tower 10 m high at the top of a flag staff, the depression angle of 30° is made. Also, looking at the bottom of the staff from the tower, the angle of the depression made is of 60°. Find the height of the flag staff.

Answer 1

In the figure, PQ is the tower of height 10 m and AB is the flagstaff.

∠MPB = ∠PBQ, ∠MPA = ∠PAS  ......(Alternate interior angles)

Now, in ΔPBQ

tan 60° = ${\displaystyle \frac{PQ}{BQ}}$

⇒ ${\displaystyle \sqrt{3}=\frac{10}{BQ}}$

⇒ BQ = ${\displaystyle \frac{10}{\sqrt{3}}}$

Also, in ΔPAS

tan 30° = ${\displaystyle \frac{PS}{AS}}$

⇒ ${\displaystyle \frac{1}{\sqrt{3}}=\frac{PS}{BQ}}$  ......[∵ BQ = AS]

⇒ ${\displaystyle \frac{1}{\sqrt{3}}=\frac{PS}{\frac{10}{\sqrt{3}}}=\frac{\sqrt{3}PS}{10}}$

⇒ ${\displaystyle \sqrt{3}\times \sqrt{3}PS}$ = 10

⇒ 3 PS = 10

⇒ PS = ${\displaystyle \frac{10}{3}}$

So, SQ = AB = PQ – PS

= ${\displaystyle 10-\frac{10}{3}}$

= ${\displaystyle \frac{3\times 10-10}{3}}$

= ${\displaystyle \frac{20}{3}}$

= 6.67

As a result, the flagstaff's height is 6.67 m.