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Arnab visited a place located at longitude 82.5°E on 21 March. He observed that at local noon, shadow of 2 meter strick standing vertically on the ground was 53.6 cm long due south.

Question : Find the day on which the shadow of this stick at the local noon will be longest and find the length and direction of that shadow.

The length of the shadow was 53.6 cm = 0.536 m

So the angle created by the stick with the line joining the head of the stick and the head of the shadow was, $\theta = tan^{-1}\frac{0.536}{2} \approx 15^{\circ}$

This is equal to the zenith angle, the angular distance from the zenith (the point directly above our/observer's head), of the sun, $z_{sun} = 15^{\circ}\quad North \qquad [\because \text{the shadow was on the south}]$

On 21 March, Sun was directly above the equator, but from Arnab's position it was 15 degree north,

so the longitude of his position is 15 degree South.

Therefore, sun will be furthest away from the location when it will further into the northern hemisphere, 21 June.

At 21 June local noon, the sun will be directly above the 23.5 degree north longitude.

So the zenith angle of the sun seen by Arnab will be, 15+23.5 = 38.5 degree.

If the length of the shadow is l, we can write, $\tan(38.5^{\circ})= \frac{l}{2} \Rightarrow l = 2\times \tan(38.5^{\circ}) = 2\times 0.80 = 1.60\, m \quad\text{due South.}$