The shadow of the pole standing on a level surface is found to be 5 m shorter when the sun's elevation is 60° than when it is 45°. What is the height of the pole?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{5\sqrt{3}}{2}(\sqrt{3}+1)\) m
The sun’s rays passing the top of the pole end at the tip of the shadow, so the pole, its shadow and the ray form a right triangle in which the angle of elevation \(\theta\) of the sun gives \(\tan\theta=\frac{\text{height}}{\text{shadow}}\). The higher sun (60°) casts the shorter shadow, \(x\); at 45° the shadow is \(x+5\).
Equation (1) (sun at 45°):
\(\tan45^\circ=\displaystyle\frac{h}{x+5}=1\)
So \(x=h-5\) (1)
Equation (2) (sun at 60°):
\(\tan60^\circ=\displaystyle\frac{h}{x}=\sqrt{3}\)
So \(h=\sqrt{3}\,x\) (2)
Finding \(h\) (putting \(x\) from (1) into (2)):
\(h=\sqrt{3}(h-5)\)
\(h(\sqrt{3}-1)=5\sqrt{3}\)
\(\displaystyle h=\frac{5\sqrt{3}}{\sqrt{3}-1}\)
Rationalising, since \((\sqrt{3}-1)(\sqrt{3}+1)=2\): \(\displaystyle h=\frac{5\sqrt{3}(\sqrt{3}+1)}{2}\) m, about 11.83 m