A tree standing on a horizontal plane is leaning towards east At two points situated at distances a and - Maths - Some Applications of Trigonometry

Question

A tree standing on a horizontal plane is leaning towards east
At two points situated at distances a and b exactly due west on it
, the angles of elevation of the top are respectively α and
β Prove that the height of the top from the ground is
(b-a)tanαtanβ/tanα-tanβ

Ankush Jain · Accepted Answer

Let PQ be the leaning tree and R and S be the two given points at distance a and b from P respectively.

Let PT = x and QT = h

In ∆PQT

$\tan θ = \frac{QT}{PT} = \frac{h}{x} \Rightarrow x = \frac{h}{\tan θ} ....... (1)$

In ∆QRT,

$\tan α = \frac{QT}{RT} = \frac{h}{x + a} ....... (2)$

Putting the value of x from (1) in (2), we get

$\tan α = \frac{h}{\frac{h}{\tan θ} + a} = \frac{h \tan θ}{h + a \tan θ}$

⇒ tan α (h + a tan θ) = h tan θ

h tan α + a tan θ tan α = h tan θ

⇒ h tan α = h tan θ – a tan θ tan α

h tan α = tan θ (h – a tan α)

$\Rightarrow \tan θ = \frac{h \tan α}{(h - a \tan α)} .........(3)$

Now, In ∆QST

$\tan β = \frac{QT}{ST} = \frac{h}{x + b} \tan β = \frac{h}{\frac{h}{\tan θ} + b} (Using (1)) \tan β = \frac{h \tan θ}{h + b \tan θ}$

Substituting the value of tan θ from (3), we get

$\tan β = \frac{h \times \frac{h \tan α}{(h - a \tan α)}}{h + \frac{b \times h \tan α}{(h - a \tan α)}} \tan β = \frac{h^{2} \tan α}{(h - a \tan α)} \times \frac{(h - a \tan α)}{(h (h - a \tan α) + b h \tan α)} \tan β = \frac{h^{2} + \tan α}{h^{2} - a h \tan α + b h \tan α}$

⇒ h² tan β – ah tan α tan β + bh tan α tan β – h² tan α = 0

⇒ h (h tan β – h tan α + b tan α tan β – a tan α tan β) = 0

⇒ h (tan β – tan α) + tan α tan β (b – a) = 0

$\Rightarrow h = \frac{- (b - a) \tan α \tan β}{\tan β - \tan α} \Rightarrow h = \frac{(b - a) \tan α \tan β}{\tan α - \tan β}$

59

A tree standing on a horizontal plane is leaning towards east. At two points situated at distances a and b exactly due west on it , the angles of elevation of the top are respectively α and β. Prove that the height of the top from the ground is (b-a)tanαtanβ/tanα-tanβ.