From a point P on a level ground, the angle of elevation of the top of a tower is $30^\circ$. If the tower is 100 m high, the distance of point P from the foot of the tower is

### Concept & Tangent Ratio This problem involves finding the adjacent side (distance from the base) of a right-angled triangle when the opposite side (height) and the angle of elevation are given. $$ \tan(\theta) = \frac{\text{Height}}{\text{Distance}} $$ ### Step-by-Step Solution 1. Given: Height of the tower ($h$) = $100$ m. Angle of elevation ($\theta$) = $30^\circ$. 2. Let the distance of point P from the foot of the tower be $x$. 3. Apply the tangent function: $$ \tan(30^\circ) = \frac{100}{x} $$ 4. Substitute the value of $\tan(30^\circ) = \frac{1}{\sqrt{3}}$: $$ \frac{1}{\sqrt{3}} = \frac{100}{x} $$ 5. Solve for $x$: $$ x = 100\sqrt{3} $$ 6. Using the approximate value of $\sqrt{3} \approx 1.732$: $$ x = 100 \times 1.732 = 173.2 \text{ m} $$ 7. The closest option is $173$ m. ### Exam Strategy & Shortcut For any $30^\circ$-$60^\circ$-$90^\circ$ triangle, the sides are in the ratio $1 : \sqrt{3} : 2$. The side opposite $30^\circ$ is $1$ part ($100$ m), so the side opposite $60^\circ$ (the distance) is $\sqrt{3}$ parts, instantly giving $100\sqrt{3}$. ### Common Pitfall Forgetting the decimal value of $\sqrt{3}$ ($\approx 1.732$) and guessing the answer, or accidentally using the sine or cosine functions instead of tangent. ### Final Answer Therefore, the correct answer is **173 m**.