Question

From a point on the ground, the angle of elevation of the top of a pedestal is 30° and that of the top of the flagstaff fixed on the pedestal is 60°. If the length of the flagstaff is 5 m, then find the height of the pedestal and its distance from the point of observation on the ground. (Use `sqrt(3)` = 1.73)

Answer 1

Let CD be the flagstaff, BD be the pedestal and A be the point on the ground from which the elevation angles are measured. 

Height of flagstaff (CD) = 5 m

Height of pedestal (BD) = h

Distance between point A and pedestal (BD) = x

Angle of elevation from point A to the top of the flagstaff = 60°

So, ∠CAB = 60°

Angle of elevation from point A to the top of the pedestal = 30°

So, ∠DAB = 30°

In right-angle triangle ABC,

tan 60° = `(BC)/(AB)`

tan 60° = `(BD + CD)/(AB)`   ...(∵ BC = BD + CD)

tan 60° = `(h + 5)/x`

`sqrt(3) = (h + 5)/x`   ...`(∵ tan 60^circ = sqrt(3))`

x = `(h + 5)/sqrt(3)`   ...(1)

In right-angle triangle ABD:

tan 30° = `(BD)/(AB)`

tan 30° = `h/x`

`1/sqrt(3) = h/x`   ...`(∵ tan 30^circ = 1/sqrt(3))`

x = `sqrt(3)h`   ...(2)

Now, putting the value of x in equation (1),

`sqrt(3)h = (h + 5)/sqrt(3)`

⇒ `sqrt(3)(sqrt(3)h) = h + 5`   ...(By cross multiplying)

⇒ 3h = h + 5

⇒ 3h – h = 5

⇒ 2h = 5

⇒ h = `5/2`

⇒ h = 2.5 m

Now, putting the value of h in equation (2),

x = `sqrt(3)h`

⇒ x = `sqrt(3) xx 2.5`

⇒ x = 1.73 × 2.5   ...`(∵ sqrt(3) = 1.73)`

x = 4.325 m

Hence, height of the pedestal (h) is 2.5 m and distance between point A and pedestal (x) is 4.325 m.