If the angle of elevation of the sun changes from 30° to 45°, the length of the shadow of a pillar decreases b

Question

If the angle of elevation of the sun changes from 30° to 45°, the length of the shadow of a pillar decreases by 80 m. The height of the pillar is:

Examoogle · Accepted Answer

The correct answer is Option A: $40(\sqrt{3} + 1) 	ext{ cm}$. Use the formula $h = \frac{d}{\cot(	heta_1) - \cot(	heta_2)}$. For $30^\circ$ and $45^\circ$, this simplifies to $h = \frac{d}{\sqrt{3} - 1} = \frac{d(\sqrt{3} + 1)}{2}$. Substituting $d = 80$, we get $40(\sqrt{3} + 1)$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the formula $h = \frac{d}{\cot(	heta_1) - \cot(	heta_2)}$. For $30^\circ$ and $45^\circ$, this simplifies to $h = \frac{d}{\sqrt{3} - 1} = \frac{d(\sqrt{3} + 1)}{2}$. Substituting $d = 80$, we get $40(\sqrt{3} + 1)$.

Formula Used: $$h = \frac{d}{\cot(	heta_1) - \cot(	heta_2)}$$

Given: Angle of elevation changes from 30 degrees to 45 degrees, decrease in shadow length d = 80 m.

Steps: Define the geometry: Let $h$ be the height of the pillar and $x$ be the length of the shadow at $45^\circ$. The shadow length at $30^\circ$ is $x + 80$.. Solve for x: Since $	an(45^\circ) = 1$ and $	an(30^\circ) = \frac{1}{\sqrt{3}}$, we have $h = x$ and $h = \frac{x+80}{\sqrt{3}}$.. Calculate the height h: Rearranging the equation $h(\sqrt{3} - 1) = 80$ gives $h = \frac{80}{\sqrt{3} - 1}$. Rationalize the denominator by multiplying by $(\sqrt{3} + 1)$.

Answer: A is correct because the calculated height of the pillar is $40(\sqrt{3} + 1)$ m.

Exam Tip: This problem is a standard application of the 'shadow' principle in trigonometry, often used in navigation and surveying calculations.

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Option B: $30(\sqrt{3} + 1) 	ext{ cm}$

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Option C: $10(\sqrt{6} + 1) 	ext{ cm}$

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Option D: $20(\sqrt{3} + 1) 	ext{ cm}$

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