GATE ECE 2018 | GA | Question: 5

Question

A $1.5$ m tall person is standing at a distance of $3$ m from a lamp post. The light from the lamp at the top of the post casts her shadow. The length of the shadow is twice her height. What is the height of the lamp post in meters?

• A. $1.5$
• B. $3$
• C. $4.5$
• D. $6$

Answer 1

As we know when light is falling top of an object it casts the shadow at behind of the object

Here, we can see from $\bigtriangleup ABE$ and $\bigtriangleup CDE$

          $ \angle B = \angle D $  &   $\angle E = \angle E$

∴ By proportionality theorem,

     $\dfrac{AB}{CD} = \dfrac{BE}{DE}$

Given,  $   CD = 1.5 \: m \\ BD = 3 \: m  \\ DE = 3 \: m \\ \therefore AB = \: ?$

∴  $\dfrac{AB}{1.5} = \dfrac{3 + 3}{3}$

   $ AB =  \dfrac{6}{3} \times 1.5$

    $AB = 3 \: m$

∴ Height of the Lamp post is $3 \: m $ which is option  B)

Comments

There are 2 more methods:

1) If $\angle E = \alpha$

$tan(\alpha) = \dfrac{CD}{DE} = \dfrac{AB}{BE}$

2) Area($\triangle ABE$) = Area($\triangle CDE$) + Area(trapezium $ABDC$)

Answer 2

As we know when light is falling top of an object it casts the shadow at behind of the object

Here, we can see in △ABE and △CDE

           ∠B = ∠D   &   ∠E = ∠E

∴ By proportionality theorem,

     $\dfrac{AB}{CD} = \dfrac{BE}{DE}$

Here,   AB =  ?

           CD = 1.5 m

           BD = 3 m

           DE = 3 m

∴  $\dfrac{AB}{1.5} = \dfrac{3 + 3}{3}$

    AB = $ \dfrac{6}{3} * 1.5$

    AB = 3 m

∴ Height of the Lamp post is 3 m option  B)