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Question: The length of the shadows of a vertical pole of height h, thrown by the sun’s ray at three different...

The length of the shadows of a vertical pole of height h, thrown by the sun’s ray at three different moments are h, 2h and 3h. The sum of the angles of elevation of the rays at these three moments is equal to

A

π2\frac { \pi } { 2 }

B

π3\frac { \pi } { 3 }

C

π4\frac { \pi } { 4 }

D

π6\frac { \pi } { 6 }

Answer

π2\frac { \pi } { 2 }

Explanation

Solution

tan⁡α=hh=1⇒α=45∘\tan \alpha = \frac { h } { h } = 1 \Rightarrow \alpha = 45 ^ { \circ }

tan⁡γ=h3h\tan \gamma = \frac { h } { 3 h }

⇒ γ=tan⁡−1(13)\gamma = \tan ^ { - 1 } \left( \frac { 1 } { 3 } \right)

∴α+β+γ=45∘+tan⁡−112+tan⁡−113\alpha + \beta + \gamma = 45 ^ { \circ } + \tan ^ { - 1 } \frac { 1 } { 2 } + \tan ^ { - 1 } \frac { 1 } { 3 } =45∘+45∘= 45 ^ { \circ } + 45 ^ { \circ } =π2= \frac { \pi } { 2 }