question_answer

When the angle of elevation of the sun increases from \[30{}^\circ \] to \[60{}^\circ ,\]the shadow of a post is diminished by 5 m. Then, the height of the post is

A) \[\frac{5\sqrt{3}}{2}\,m\]

B) \[\frac{2\sqrt{3}}{5}\,m\]

C) \[\frac{2}{5\sqrt{3}}\,m\]

D) \[\frac{4}{5\sqrt{3}}\,m\]

Correct Answer: A

Solution :

[a] Let the height of the post \[=h\,\,m\] In \[\Delta ACB,\] \[\tan 30{}^\circ =\frac{h}{5+x}\]                     (i) In\[\Delta DAB,\]\[\tan 60{}^\circ =\frac{h}{x}\]\[\Rightarrow \]\[x=\frac{h}{\sqrt{3}}\] From Eq. (i), we get \[\frac{1}{\sqrt{3}}=\frac{h}{5+\frac{h}{\sqrt{3}}}\] \[\Rightarrow \]   \[\frac{5}{\sqrt{3}}+\frac{h}{3}=h\]\[\Rightarrow \]\[h-\frac{h}{3}=\frac{5}{\sqrt{3}}\] \[\therefore \]      \[h=\frac{5\times 3}{2\times \sqrt{3}}=\frac{5\sqrt{3}}{2}\,m\]