Question

A 1.5 m tall boy stands at a distance of 3 m from
the lamp-post and casts a shadow of 4.5 m on the
ground. Find the height of the lamp-post.

(answer it with a picture please♥)

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Answer 1

Question : -

A 1.5 m tall boy stands at a distance of 3 m from the lamp-post and casts a shadow of 4.5 m on the ground. Find the height of the lamp-post.

Answer : -

Given : -

A 1.5 m tall boy stands at a distance of 3 m from the lamp-post and casts a shadow of 4.5 m on the ground.

Required to find : -

• Height of the lamp post ?

Trigonometric ratio used : -

$\boxed{\sf{ \tan \theta = \dfrac{ Opposite \; Side }{ Adjacent \; Side } }}$

Solution : -

Let's draw the diagram for the given data ;

- - - : Diagram : - - -

$\setlength{\unitlength}{20} \begin{picture}(6,6) \linethickness{2.5} \put(1,1){\line(1,0){10}}\put(11,1){\line(0,1){5}}\put(6,1){\line(0,1){2.5}}\qbezier(1,1)(1,1)(11,6)\qbezier(2,1.5)(3,1.3)(2,1)\put(2.8,1.25){ \bf{ \theta} }\put(3,0.5){ \bf 4.5 \; m }\put(8,0.5){ \bf 3 \; m }\put(6.15,2){ \bf 1.5 \; m }\put(0.8,0.5){ \bf E }\put(6,0.5){ \bf C }\put(5.8,3.8){ \bf D }\put(11,0.5){ \bf B }\put(11,6.15){ \bf A }\end{picture}$

Here,

From the diagram we can conclude that ;

AB = Height of the lamp post

BC = Distance between the lamp-post and boy = 3 m

CD = Height of the boy = 1.5 m

CE = Length of shadow casted = 4.5 cm

Now,

Let's find the height of the lamp post ?

So,

let's consider ∆ CDE

In which ,

∠DCE = 90° , ∠CED = Θ

According to problem ;

Tan Θ = Opposite side/ Adjacent side

since,

• Opposite side = CD

• Adjacent side = CE

Tan Θ = CD/CE $\longrightarrow{\tt{\bf{ Equation - 1 }}}$

Consider this as equation - 1

Similarly,

Now,

Consider ∆ ABE

In ∆ ABE , in which

∠ABE = 90° , ∠BEA = Θ

According to problem ;

Tan Θ = Opposite side/ Adjacent side

Since,

• Opposite side = AB

• Adjacent side = BE

Tan Θ = AB / BE $\longrightarrow{\tt{\bf{ Equation - 2 }}}$

Consider this as equation - 2

Consider equation 1 & equation 2

➜ Tan Θ = CD/CE

➜ Tan Θ = AB/BE

Since,

The LHS part is equal let's equate the RHS part .

CD/CE = AB/BE

However,

• CD = 1.5 cm

• CE = 4.5 cm

• BC = 3 cm

This implies ;

➜ 1.5/4.5 = AB/BE

Since, BE = BC + CE

➜ 1.5/4.5 = AB/BC + CE

➜ 1.5/4.5 = AB/ 4.5 + 3

➜ 1.5/4.5 = AB/7.5

By cross multiplication

➜ 4.5 x AB = 7.5 x 1.5

➜ AB = 7.5 x 1.5/ 4.5

➜ AB = 11.25/4.5

➜ AB = 112.5/45

➜ AB = 2.5 meters

( Since, AB = Height of the lamp post )

Therefore,

Height of the lamp post = 2.5 meters