Question

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

• A. ${45}^0$
• B. ${35}^0$
• C. ${75}^0$
• D. ${60}^0$

Answer 1

Answer: (C)

${75}^0$

$RS=C$

$RQ=C$

Using $m-n$ theorem, we get

$\Rightarrow (C+C)\cot (\theta -30°)=C\cot 15°-C\cot 30°$

$\Rightarrow 2C\cot (\theta -30°)=C[\frac{\cos 15°}{\sin 15°}-\frac{\cos 30°}{\sin 30°}]$

$\Rightarrow \cot (\theta -30°)=\frac{1}{2}\left[\frac{\sin 30°\cos 15°-\cos 30°\sin 15°}{\sin 30°\sin 15°}\right]$

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

$\Rightarrow \cot (\theta -30°)=\frac{1}{2\times \frac{1}{2}}=1=\cot 45°$

$\Rightarrow \cot (\theta -30°)=\cot 45°$

$\Rightarrow \theta -30°=45°$

$\Rightarrow \theta =75°$

Hence, the answer is $75°.$