The angle of elevation of the Sun, when the length of the shadow of a tree is \(\sqrt{3}\) times the height of the tree, is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 30°
The tree, its shadow and the Sun's ray from the top of the tree to the end of the shadow form a right-angled triangle. The angle of elevation \(\theta\) is at the end of the shadow, so \(\tan\theta=\) height \(\div\) shadow.
Let the height of the tree be \(h\); then the shadow is \(\sqrt{3}\,h\)
Opposite side over adjacent side: \(\tan\theta=\displaystyle\frac{h}{\sqrt{3}\,h}=\frac{1}{\sqrt{3}}\)
The angle whose tangent is \(\displaystyle\frac{1}{\sqrt{3}}\) is \(30^\circ\), so \(\theta=30^\circ\)
Tip: A shadow \(\sqrt{3}\) times the height means a low Sun, 30°; a height \(\sqrt{3}\) times the shadow means a high Sun, 60°; a shadow equal to the height means 45°.