Question

A flagstaff stands on a vertical tower. At a distance 10m from the base of tower. Thetower and the flagstaff makes angles of 45degree and 15degree.find the length of the flagstaff.

Answer 1

Let AD be the flagstaff and CD be the building.

Assume that the flagstaff and building subtend equal angles at point B.
Given that AD = 50 m, CD = h and BC = 200 m

Let ABD = θ, DBC = θ     (∵ flagstaff and building subtend equal angles at a point on level ground).
Then, ABC = 2θ

From the right  BCD, 
tanθ=DCBC=h200⋯(eq:1)tan⁡θ=DCBC=h200⋯(eq:1)

From the right  BCA, 
tan2θ=ACBC=AD + DC200=50 + h200tan⁡2θ=ACBC=AD + DC200=50 + h200

⇒2tanθ1−tan2θ=50 + h200⇒2tan⁡θ1−tan2⁡θ=50 + h200 (∵tan(2θ)=2tanθ1−tan2θ)(∵tan⁡(2θ)=2tan⁡θ1−tan2⁡θ)

⇒2(h200)1−h22002=50 + h200⇒2(h200)1−h22002=50 + h200(∵ substituted value of tan θ from eq:1)

⇒2h=(1−h22002)(50 + h)⇒2h=50+h−50h22002−h32002⇒2h=(1−h22002)(50 + h)⇒2h=50+h−50h22002−h32002

⇒2(2002)h⇒2(2002)h =50(200)2+h(200)2−50h2−h3=50(200)2+h(200)2−50h2−h3 (∵ multiplied LHS and RHS by 2002)

⇒h3+50h2+(200)2h−50(200)2=0