Question
Mohan looks at a tree top and the angle made is 45°. He moves 10 cm back and again looks at the tree top but this time angle made is 30°. How high is the tree top from ground?
Answer: Option D
Let PQ be three and M and N be positions where Mphan stands.
$$\eqalign{
& {\text{Now,}}\,{\text{tan}}{45^ \circ } = 1 = \frac{{PQ}}{{MQ}} \cr
& \therefore PQ = MQ \cr
& \tan {30^ \circ } = \frac{1}{{\sqrt 3 }} = \frac{{PQ}}{{NQ}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{PQ}}{{10 + MQ}} \cr
& \therefore 10 + MQ = \sqrt 3 \,PQ \cr} $$
$$\therefore 10 + PQ = \sqrt 3 \,PQ$$ (As, PQ = MQ)
$$\therefore PQ = \frac{{10}}{{\sqrt 3 - 1}}\,{\text{cm}}$$
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Let PQ be three and M and N be positions where Mphan stands.
$$\eqalign{
& {\text{Now,}}\,{\text{tan}}{45^ \circ } = 1 = \frac{{PQ}}{{MQ}} \cr
& \therefore PQ = MQ \cr
& \tan {30^ \circ } = \frac{1}{{\sqrt 3 }} = \frac{{PQ}}{{NQ}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{PQ}}{{10 + MQ}} \cr
& \therefore 10 + MQ = \sqrt 3 \,PQ \cr} $$
$$\therefore 10 + PQ = \sqrt 3 \,PQ$$ (As, PQ = MQ)
$$\therefore PQ = \frac{{10}}{{\sqrt 3 - 1}}\,{\text{cm}}$$
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