Question
The angle of elevation of a stationary cloud from a point $$2500\,m$$ above a lake is $${15^ \circ }$$ and the angle of depression of its reflection in the lake is $${45^ \circ }.$$ The height of cloud above the lake level is
A.
$$2500\sqrt 3 \,{\text{metres}}$$
B.
$$2500 \,{\text{metres}}$$
C.
$$500\sqrt 3 \,{\text{metres}}$$
D.
None of these
Answer :
$$2500\sqrt 3 \,{\text{metres}}$$
Solution :
$$\eqalign{ & \left( {H - h} \right)\cot {15^ \circ } = \left( {H + h} \right)\cot {45^ \circ } \cr & {\text{or }}H = \frac{{h\left( {\cot {{15}^ \circ } + 1} \right)}}{{\left( {\cot {{15}^ \circ } - 1} \right)}} \cr} $$
Since $$h = 2500$$ and substitute
$$\cot {15^ \circ } = 2 + \sqrt 3 ,{\text{ we get, }}H = 2500\sqrt 3 $$
$$\eqalign{ & \left( {H - h} \right)\cot {15^ \circ } = \left( {H + h} \right)\cot {45^ \circ } \cr & {\text{or }}H = \frac{{h\left( {\cot {{15}^ \circ } + 1} \right)}}{{\left( {\cot {{15}^ \circ } - 1} \right)}} \cr} $$
Since $$h = 2500$$ and substitute
$$\cot {15^ \circ } = 2 + \sqrt 3 ,{\text{ we get, }}H = 2500\sqrt 3 $$