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° = `(PQ)/(BQ)`

⇒ `sqrt(3) = 10/(BQ)`

⇒ BQ = `10/sqrt(3)`

Also, in ΔPAS

tan 30° = `(PS)/(AS)`

⇒ `1/sqrt(3) = (PS)/(BQ)`  ......[∵ BQ = AS]

⇒ `1/sqrt(3) = (PS)/(10/sqrt(3)) = (sqrt(3) PS)/10`

⇒ `sqrt(3) xx sqrt(3) PS` = 10

⇒ 3 PS = 10

⇒ PS = `10/3`

So, SQ = AB = PQ – PS

= `10 - 10/3`

= `(3 xx 10 - 10)/3`

= `20/3`

= 6.67

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