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The shadow of a pole standing on a horizontal plane is a metre

Question

The shadow of a pole standing on a horizontal plane is $a$ metres longer when the sun's elevation is $θ$ than when it is $ϕ$ . The height of the pole will be

a \frac{cos θ \cdot cos ϕ}{cos ( θ - ϕ )}

metre

a \frac{sin θ sin ϕ}{sin ( ϕ - θ )}

metre

a \frac{sin θ cos ϕ}{sin ( θ - ϕ )}

metre

a \frac{sin ϕ cos θ}{cos ( θ - ϕ )}

metre

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The shadow of a pole standing on a horizontal plane is

a

metres longer when the sun's elevation is

θ

than when it is

ϕ

. The height of the pole will be

12 mins ago

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12 mins ago

Text solutionVerified

Let the height of the poie standing on a horizontal plane be

h

Let the length of the shadow be

x

and

(x + a)

, when the sun's elevation is

ϕ

and

θ

respectively.

∵ \frac{x}{h} = cot ϕ

\Rightarrow x = h cot ϕ

....(i)

\frac{x + a}{h} = cot θ

\Rightarrow (x + a) = h cot θ

....(ii)

\Rightarrow (x + a) - x = h cot θ - h cot ϕ

\Rightarrow a = h (co tθ - co tϕ)

\Rightarrow h = \frac{a}{( \frac{cos θ}{sin θ} - \frac{cos ϕ}{sin ϕ} )}

\Rightarrow h = a \frac{sin θ - sin ϕ}{( sin ϕ cos θ - cos ϕ sin θ )}

\Rightarrow h = a \frac{sin θ sin ϕ}{sin ( ϕ - θ )}

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Question Text	The shadow of a pole standing on a horizontal plane is $a$ metres longer when the sun's elevation is $θ$ than when it is $ϕ$ . The height of the pole will be
Answer Type	Text solution:1
Upvotes	151