### XAT 2011 Question 74

Instructions

Based on the following information

A man standing on a boat south of a light house observes his shadow to be 24 meters long, as measured at the sea level. On sailing 300 meters eastwards, he finds his shadow as 30 meters long, measured in a similar manner. The height of the man is 6 meters above sea level.

Question 74

# The height of the light house above the sea level is:

Solution

KL = lighthouse

BA = initial position man of man and BC = shadow

After moving 300 m east, DE = new position of man and EF = shadow

Given : AB = DE = 6 m

BC = 24 m and EF = 30 m and BE = 300 m

$$\triangle$$ LBE is right angled triangle (sea level).

To find : KL = ?

Solution : In $$\triangle$$ KLF and $$\triangle$$ DEF

=> $$\angle KLF = \angle DEF = 90$$

$$\angle KFL = \angle DFE$$ (common angle)

=> $$\triangle KLF \sim \triangle DEF$$

=> $$\frac{KL}{DE} = \frac{LF}{EF}$$ -----------Eqn(I)

Similarly, $$\triangle KLC \sim \triangle ABC$$

=> $$\frac{KL}{AB} = \frac{LC}{BC}$$ ----------Eqn(II)

From eqn (I) and (II), and using AB = DE

=> $$\frac{LC}{BC} = \frac{LF}{EF}$$

=> $$\frac{LC}{24} = \frac{LF}{30}$$

=> $$\frac{LC}{LF} = \frac{24}{30} = \frac{4}{5}$$

If, LC is 4 part $$\equiv$$ LF is 5 part

=> $$LB = 4x$$ and $$LE = 5x$$

$$\because$$ $$\triangle$$ LBE is right angled triangle

=> $$(LE)^2 - (LB)^2 = (BE)^2$$

=> $$25X^2 - 16X^2 = 90000$$

=> $$x^2 = \frac{90000}{9} = 10000$$

=> $$x = \sqrt{10000} = 100$$

=> $$LB = 400$$ and $$LE = 500$$

=> $$LC = LB + BC = 400 + 24 = 424$$

Now, using Eqn (II), we get :

=> $$KL = \frac{LC}{BC} \times AB$$

= $$\frac{424}{24} \times 6 = \frac{424}{4}$$

= $$106 m$$