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A man 1.8 m tall stands at a distance of 3.6 m from a lamp post and casts a shadow of 5.4 m on the ground. Find the height of the lamp post.

Heights & Distances

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Answer

Given,

A man 1.8 m tall stands at a distance of 3.6 m from a lamp post and casts a shadow of 5.4 m on the ground. Find the height of the lamp post. Volume And Surface Area of solid RSA Mathematics Solutions ICSE Class 10.

AB is the lamp post and CD is the height of man and CE is the shadow of the man.

CE || DB.

Take AB = x and CD = 1.8 m

FA = CD = 1.8 m

DF = CA = 3.6 m

BF = AB - FA = (x - 1.8) m

Shadow (EC) = 5.4 m

Considering right angled △BDF, we get :

⇒tan⁡θ=BFDF⇒tan⁡θ=x−1.83.6 ….(1)\Rightarrow \tan \theta = \dfrac{BF}{DF} \\[1em] \Rightarrow \tan \theta = \dfrac{x - 1.8}{3.6} \text{ ….(1)}

Considering right angled △DCE, we get :

⇒tan⁡θ=CDEC⇒tan⁡θ=1.85.4=13 ….(2)\Rightarrow \tan \theta = \dfrac{CD}{EC} \\[1em] \Rightarrow \tan \theta = \dfrac{1.8}{5.4} = \dfrac{1}{3} \text{ ….(2)}

Comparing Eq 1 and Eq 2 we get,

⇒x−1.83.6=13⇒3x−5.4=3.6⇒3x=5.4+3.6⇒3x=9⇒x=3.\Rightarrow \dfrac{x - 1.8}{3.6} = \dfrac{1}{3} \\[1em] \Rightarrow 3x - 5.4 = 3.6 \\[1em] \Rightarrow 3x = 5.4 + 3.6 \\[1em] \Rightarrow 3x = 9 \\[1em] \Rightarrow x = 3.

Hence, the height of the lamp post is 3 meters.

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