[h3 + 50h2...] A flagstaff is placed on top of a building. The flagstaff and building subtend equal angles at a poi

A flagstaff is placed on top of a building. The flagstaff and building subtend equal angles at a point on level ground which is 200 m away from the foot of the building. If the height of the flagstaff is 50 m and the height of the building is h, which of the following is true?

h³ - 50h² + (200)²h + (200)²50 = 0

None of these

h³ + 50h² + (200)²h - (200)²50 = 0

h³ - 50h² - (200)²h + (200)²50 = 0

Explanation:

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

From the right $\triangle$ BCD,
tanθ=$\dfrac{DC}{BC}=\dfrac{h}{200}$ ⋯(eq:1)

From the right $\triangle$ BCA,
tan2θ=$\dfrac{AC}{BC}=\dfrac{AD + DC}{200}=\dfrac{50 + h}{200}$

⇒$\dfrac{2tanθ}{1−tan2θ}=\dfrac{50 + h}{200}$ (∵tan(2θ)=$\dfrac{2tanθ}{1−tan2θ}$)

⇒$\dfrac{2\left( \dfrac{h}{200}\right)}{1−\dfrac{h^{2}}{200^{2}}}=\dfrac{50 + h}{200}$(∵ substituted value of tan θ from eq:1)

⇒2h=$\left(1−\dfrac{h^{2}}{200^{2}}\right) $ (50 + h)
⇒2h=50+h−$\dfrac{50h^{2}}{200^{2}}−\dfrac{h^{3}}{200^{2}}$

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

⇒h³+50h²+(200)2h−50(200)²=0

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