Question : If the angle of elevation of the sun changes from 45° and 65°, then the length of the shadow of a pillar decreases by 10 m. The height of the pillar is:
Option 1: $5\left (3-\sqrt{3} \right)$ m
Option 2: $5\left (\sqrt{3}+1 \right)$ m
Option 3: $15\left (\sqrt{3}+1 \right)$ m
Option 4: $5\left (3+\sqrt{3} \right)$ m
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Correct Answer: $5\left (3+\sqrt{3} \right)$ m
Solution : Let the height of pillar = $AB$ = $h$ m Given: $CD=10$ m $\angle ACB = 45^{\circ} \quad \angle ADB = 60^{\circ}$ Let $BD = x$ m From $\Delta ABC$ $\tan 45^{\circ} = \frac{AB}{BC}$ $\Rightarrow 1=\frac{h}{x + 10}$ $\Rightarrow h = (x + 10)$ m . . . . . . .$(i)$ From $\Delta ABD$ $\tan 60^{\circ} = \frac{AB}{BD}$ $\Rightarrow\sqrt{3} = \frac{h}{x}$ $\Rightarrow x =\frac{h}{\sqrt{3}}$ m . . . . .$(ii)$ From equation $(i)$ $\Rightarrow h = \frac{h}{\sqrt{3}} + 10$ $\Rightarrow h - \frac{h}{\sqrt{3}} = 10$ $\Rightarrow h\left(\sqrt{3} - 1\right)=10\sqrt{3}$ $\Rightarrow h = \frac{10\sqrt{3}}{\sqrt{3}-1}$ $\Rightarrow h = \frac{10\sqrt{3}(\sqrt{3}+1)}{(\sqrt{3}-1)(\sqrt{3}+1)} $ $\Rightarrow h = \frac{10\sqrt{3}(\sqrt{3} + 1)}{3 - 1}$ $\Rightarrow h = 5(3 + \sqrt{3})$ m Hence, the correct answer is $5(3 + \sqrt{3})$ m.
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Question : The angle of elevation of the sun changes from 30° to 45°. The length of the shadow of a pole decreases by 4 metres and the height of the pole is: (Assume $\sqrt3$ = 1.732)
Question : If a pole of 12 m height casts a shadow of $4\sqrt{3}$ m long on the ground, then the sun's angle of elevation at that instant is:
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:
Question : The ratio of the length of a rod and its shadow is $1: \sqrt{3}$. The angle of elevation of the sun is:
Question : If $a+b :\sqrt{ab} = 4:1 $ where $ a > b > 0$, then $ a:b$ is:
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