Question

From the top of a hill the angle of depression of two consecutive mile stones on east are ${30}^o$ and ${45}^o$. The height of the hill is _____ km.

• A. $(\sqrt{3}+1)$
• B. $(\sqrt{3}-1)$
• C. $\frac{1}{2}(\sqrt{3}+1)$
• D. $\frac{1}{2}(\sqrt{3}-1)$

Answer 1

The correct option is C $\frac{1}{2}(\sqrt{3}+1)$

From the given figure, AB is height of the hill $=h$ m
AD and AC are distance between the base of hill and two milestone being $x$ km & $x+1$ km respectively.

Also $\angle D$ = 45${}^o$ = $\frac{AB}{AD}\Rightarrow 1=\frac{h}{x}\Rightarrow$ $h=x$ ...eq. (1)

tan 30${}^o$ = $\frac{1}{\sqrt{3}}\Rightarrow \frac{h}{h+1}\Rightarrow$ $\sqrt{3}h=h+1$

$\Rightarrow$($\sqrt{3}-1)h$ = 1$\Rightarrow$ h = $\frac{1}{\sqrt{3}-1}=\frac{\sqrt{3}+1}{(\sqrt{3}-1)(\sqrt{3}+1)}=\frac{1}{2}(\sqrt{3}+1)$

$\therefore$ Height of the hill is $\frac{1}{2}(\sqrt{3}$ + 1) km

So, option C is correct.