Question

If the length of the shadow of a tower is $\sqrt{3}$ times that of its height, then altitude of the Sun is :

• A. $45^\circ$
• B. $30^\circ$
• C. $60^\circ$
• D. $15^\circ$

Hint:

Remember these two critical standard cases for tower shadow problems:
1. If shadow length $=$ height $\implies \tan(\theta) = 1 \implies \theta = 45^\circ$.
2. If shadow length $= \sqrt{3} \times$ height $\implies \tan(\theta) = \frac{1}{\sqrt{3}} \implies \theta = 30^\circ$.
3. If shadow length $= \frac{1}{\sqrt{3}} \times$ height $\implies \tan(\theta) = \sqrt{3} \implies \theta = 60^\circ$.
Memorizing these three scenarios allows you to solve height-and-distance multiple-choice questions instantly!

Answer 1

The Correct Option is B

Step 1: Use the cotangent ratio directly, instead of the tangent ratio.
Let $h$ be the height of the tower and $s$ be the length of its shadow, with $\theta$ the altitude of the sun (the angle of elevation from the tip of the shadow to the top of the tower).
The cotangent of $\theta$ is the ratio of the adjacent side to the opposite side in the right triangle formed by the tower, its shadow, and the sun's ray:
\[ \cot\theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{s}{h} \]

Step 2: Substitute the given relationship between shadow length and height.
We are told $s = \sqrt{3}\,h$, so:
\[ \cot\theta = \frac{\sqrt{3}\,h}{h} = \sqrt{3} \]

Step 3: Match this to the standard cotangent value.
From the standard trigonometric table, we know:
\[ \cot(30^\circ) = \sqrt{3} \]
Comparing, we get:
\[ \theta = 30^\circ \]

Step 4: Cross-check using the reciprocal relation.
Since $\cot\theta$ and $\tan\theta$ are reciprocals of each other, $\tan\theta = \frac{1}{\sqrt{3}}$, which is indeed the standard value for $\tan(30^\circ)$. This confirms our answer is consistent.

Final Answer:
The altitude of the sun is $30^\circ$, which matches Option (B). \[ \boxed{30^\circ} \]