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: Understanding the Question:
This question is based on "Some Applications of Trigonometry" (commonly known as Heights and Distances).
We are given a relationship between the height of a vertical tower and the length of the shadow it casts on the ground.
We need to find the altitude of the Sun, which is geometrically represented as the angle of elevation of the Sun from the tip of the shadow.

Step 2: Key Formula or Approach:
We model this physical situation using a right-angled triangle:

• The vertical side of the triangle represents the height of the tower ($h$).

• The horizontal ground side represents the length of the shadow ($s$).

• The angle of elevation is $\theta$, representing the altitude of the Sun.

We use the trigonometric tangent ratio, which relates the opposite side (height) to the adjacent side (shadow length) of the right-angled triangle:
\[ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{h}{s} \]

Step 3: Detailed Explanation:

• Let the height of the vertical tower be $h$.

• Let the length of the shadow cast by the tower be $s$.

• According to the problem statement, the length of the shadow is $\sqrt{3}$ times the height of the tower:
\[ s = \sqrt{3} \cdot h \]

• Let $\theta$ be the angle representing the altitude of the Sun.

• In our right-angled triangle model, apply the tangent trigonometric ratio:
\[ \tan(\theta) = \frac{\text{Height of the tower}}{\text{Length of the shadow}} = \frac{h}{s} \]

• Substitute the expression for $s$ into the equation:
\[ \tan(\theta) = \frac{h}{\sqrt{3}h} \]

• Cancel the common factor $h$ from the numerator and the denominator (since $h \neq 0$):
\[ \tan(\theta) = \frac{1}{\sqrt{3}} \]

• From our standard trigonometric ratio values, we know that:
\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

• Comparing the two sides, we get:
\[ \theta = 30^\circ \]

• Therefore, the altitude of the Sun is $30^\circ$.


Step 4: Final Answer:
The altitude of the Sun is $30^\circ$, which corresponds to Option (B).