Question : The tower is 50 metres high, its shadow is $x$ metres shorter when the sun's elevation is $45°$ than when it is $30°$. The value of $x$ (in metres) is:
Option 1: $50\sqrt{3}$
Option 2: $50\left ({\sqrt3-1} \right)$
Option 3: $50\left ({\sqrt3+1} \right)$
Option 4: $50$
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Correct Answer: $50\left ({\sqrt3-1} \right)$
Solution :
Let the height of the tower be $BC$, the shadow of the tower when the sun’s altitude is 30° be $AB$ and the shadow of the tower when the sun’s elevation is 45° be $DB$.
Given: $BC$ = 50 m
And $AB - BD = x$ m
⇒ $AD$ = $x$ m (see figure)
Now in right angle triangle $\triangle ABC$,
$\tan\angle CAB=\frac{BC}{AB}$
⇒ $\tan 30^{\circ} = \frac{50}{AB}$
⇒ $AB = \frac{50}{\tan 30^{\circ}} = 50\sqrt{3}$ m
Now in right angle triangle $\triangle CBD$,
$\tan\angle CDB = \frac{BC}{BD}$
⇒ $\tan45^\circ = \frac{50}{BD}$
⇒ $BD = 50$ m
Now, $AD = AB - BD = 50\sqrt3-50$
⇒ $AD =50(\sqrt3 - 1)$ m
Hence, the correct answer is $50(\sqrt3 - 1)$.
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