Question

A street light bulb is fixed on a pole 6 m above the level of the street. If a woman of height 1.5m casts a shadow 3 m, find how far she is away from the base of the pole?

Answer 1

Hint: In this particular question first draw the pictorial representation of the give problem it will give us a clear picture of what we have to find out later on in the solution use the concept of similar triangle that if two triangles are similar then the ratio of their respective sides are equal, so us these concepts to reach the solution of the question.

Complete step-by-step answer:

Given data:
A street light bulb is fixed on a pole 6 m above the level of the street.
Let AB be the pole of 6m in height and at top of this pole there is a street light bulb.
So at A there is a street light bulb.
Let the woman stand at x meter from the pole.
Let women be standing at point C, so BC = x meter as shown in the above figure.
It is given that the height of the women is 1.5m
Now it is also given that the woman casts a shadow of 3 m length, as shown in the figure.
Let shadow lasts till D, as shown in the figure.
Now join A and D as shown in the figure so that it touches the women at point E, as shown.
Now in triangle ABD and in triangle ECD we have,
$\angle ADB = \angle EDC$ (Common angle)
$\angle ABD = \angle ECD = {90^o}$
$\angle BAD = \angle CED$ (Corresponding angles)
So triangle ABD and triangle ECD are similar.
Now we all know that if two triangles are similar then the ratio of their respective side lengths are equal.
Therefore, $\dfrac{{AB}}{{EC}} = \dfrac{{BD}}{{CD}} = \dfrac{{AD}}{{ED}}$
$ \Rightarrow \dfrac{{AB}}{{EC}} = \dfrac{{BD}}{{CD}}$
Now BD = BC + CD so we have,
\[ \Rightarrow \dfrac{{AB}}{{EC}} = \dfrac{{BC + CD}}{{CD}}\]
No substitute the values we have,
\[ \Rightarrow \dfrac{6}{{1.5}} = \dfrac{{x + 3}}{3}\]
Now simplify we have,
\[ \Rightarrow \dfrac{{6\left( 3 \right)}}{{1.5}} = x + 3\]
\[ \Rightarrow 12 = x + 3\]
\[ \Rightarrow x = 12 - 3 = 9\] m.
So the women were standing at 9 m from the pole.
So this is the required answer.

Note: Whenever we face such types of questions always recall that when two lines are parallel to each other and a third line cuts these two lines than the angle made by the third line on these two lines are equal and called as corresponding angles, and always remember that in similar triangle the ratio of their respective sides are equal, $\dfrac{{AB}}{{EC}} = \dfrac{{BD}}{{CD}} = \dfrac{{AD}}{{ED}}$.