A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan⍺/(tanꞵ - tan𝛼)]

Solution:

Given, a vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h.

The angle of elevation of the bottom and the top of the flag staff are α and β.

We have to prove that the height of the tower is [h tan⍺/(tanꞵ - tan𝛼)]

Let FP be the flag staff and PO be the vertical tower.

Given, height of vertical staff, FP = h units

Let the height of the tower be H units

Angle of elevation, ∠PRO = ⍺

Angle of depression, ∠FRO = ꞵ

Let the distance from the point of observation to the foot of the tower, RO be x units.

In triangle PRO,

tan ⍺ = PO/RO

tan ⍺ = H/x

x = H/tan ⍺ ------------------- (1)

In triangle FRO,

tan ꞵ = FO/RO

From the figure, FO = FP + PO

FO = h + H

So, tan ꞵ = (h + H)/x

x = (h + H)/tan ꞵ ----------------- (2)

From (1) and (2),

H/tan ⍺ = (h + H)/tan ꞵ

On cross multiplication,

H(tan ꞵ) = (h+H) tan⍺

H(tan ꞵ) = H(tan ⍺) + h(tan⍺)

On solving for H,

H(tan ꞵ) - H(tan ⍺) = h(tan⍺)

H(tan ꞵ - tan ⍺) = h(tan ⍺)

H = h(tan ⍺) / (tan ꞵ - tan ⍺)

Therefore, the height of the tower is [h tan⍺/(tanꞵ - tan𝛼)]

✦ Try This: A vertical tower stands on a horizontal plane and is surmounted by a flag-staff of height 7 m. From a point on the plane, the angle of elevation of the bottom of the flag-staff is 30° and that of the top of the flag-staff is 45°. Find the height of the tower.

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 8

---

NCERT Exemplar Class 10 Maths Exercise 8.4 Problem 8

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. Prove that the height of the tower is [h tan⍺/(tanꞵ - tan𝛼)]

Summary:

A vertical tower stands on a horizontal plane and is surmounted by a vertical flag staff of height h. At a point on the plane, the angles of elevation of the bottom and the top of the flag staff are α and β, respectively. It is proven that the height of the tower is [h tan⍺/(tanꞵ - tan𝛼)]

---

☛ Related Questions:

• If tanθ + secθ = l, then prove that secθ = l2+1/2l
• If sinθ + cosθ = p and secθ + cosecθ = q, then prove that q (p2 – 1) = 2p
• If a sinθ + b cosθ = c, then prove that a cosθ – b sinθ = √(a2+b2-c2)