A conical tent has 60° angle at the vertex. The ratio of the radius and slant height is :

Question

Accepted Answer

To determine the ratio of the radius to the slant height of a conical tent with a 60° angle at the vertex, we need to consider the geometric properties of a cone.

1. **Understanding the Geometry:**
   - A cone with a vertex angle of 60° can be visualized as having an equilateral triangle when a vertical cross-section through the apex and the center of the base is taken.
   - In this scenario, the slant height (l), radius (r), and height (h) of the cone form a right triangle, where the slant height is the hypotenuse.

2. **Using Trigonometry:**
   - The vertex angle of 60° implies that each half of the angle is 30° when considering the right triangle formed by the radius, height, and slant height.
   - In a 30°-60°-90° triangle, the sides are in the ratio 1 : √3 : 2. Specifically, the side opposite the 30° angle (radius, r) is half the hypotenuse (slant height, l).

3. **Calculating the Ratio:**
   - Given the properties of a 30°-60°-90° triangle, the radius (r) is half of the slant height (l).
   - Therefore, the ratio of the radius to the slant height is 1 : 2.

4. **Justification for Option 1:**
   - Option 1 states the ratio as 1 : 2, which aligns perfectly with the geometric and trigonometric analysis of the cone with a 60° vertex angle.

5. **Explanation for Incorrectness of Other Options:**
   - **Option 2 (1 : 3):** This ratio does not correspond to the properties of a 30°-60°-90° triangle, where the radius is half the slant height.
   - **Option 3 ($1 : \sqrt{2}$):** This ratio is not applicable to a 30°-60°-90° triangle, as it does not reflect the correct relationship between the sides.
   - **Option 4 ($1 : \sqrt{3}$):** This ratio is incorrect because, in a 30°-60°-90° triangle, the side opposite the 30° angle (radius) is not related to the slant height by √3.

In conclusion, the correct ratio of the radius to the slant height for a conical tent with a 60° vertex angle is indeed 1 : 2, as stated in Option 1. This is supported by the geometric properties of the cone and the specific characteristics of a 30°-60°-90° triangle.