Question

A man observes that when he moves up a distance $c$ meters on a slope, the angle of depression of a point on the horizontal plane from the base of the slope is ${30}^0$, and when he moves up further a distance $c$ meters the angle of depression of that is ${45}^0$. The angle of inclination of the slope with the horizontal is

• A. ${60}^0$
• B. ${45}^0$
• C. ${75}^0$
• D. ${30}^0$

Answer 1

The correct option is C ${75}^0$

By $m-n$ theorem:

$(c+c)\cot (x-30)=c\times \cot 15-c\times \cot 30$

$\Rightarrow (c+c)\cot (x-30)=\frac{\cos 15\sin 30-\cos 30\sin 15}{2\sin 15\sin 30}$

$\Rightarrow \cot (x-30)=\frac{\sin (30-15)}{2\sin 15\sin 30}$

$\Rightarrow \cot (x-30)=\frac{1}{2\sin 30}$

$\Rightarrow \cot (x-30)=1$

$\Rightarrow \cot (x-30)=\cot 45$

$\Rightarrow x-30=45$

$\Rightarrow x=75$