A conical tent has `60^(@)` angle at the vertex. The ratio of its radius and slant height is :

Options

A. ` sqrt3:2`
B. `1:2`
C. `1:3`
D. can't be determined

Answer

To find the ratio of the radius to the slant height of a conical tent with a vertex angle of \(60^\circ\), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Geometry of the Cone:**
- The cone has a vertex angle of \(60^\circ\).
- When we draw a vertical line from the vertex to the base, it divides the vertex angle into two equal angles of \(30^\circ\) each.

2. **Label the Components:**
- Let \(O\) be the vertex of the cone.
- Let \(OB\) be the radius \(R\) of the base of the cone.
- Let \(CO\) be the height \(H\) of the cone.
- Let \(BC\) be the slant height \(L\) of the cone.

3. **Identify the Right Triangle:**
- In triangle \(BOC\), we have:
- Angle \(BOC = 30^\circ\) (since the vertex angle is \(60^\circ\) and is divided into two equal parts).
- \(OB = R\) (radius).
- \(BC = L\) (slant height).
- \(CO = H\) (height).

4. **Use Trigonometric Ratios:**
- In the right triangle \(BOC\), we can use the sine function:
\[ \sin(30^\circ) = \frac{R}{L} \]
- We know that \(\sin(30^\circ) = \frac{1}{2}\).

5. **Set Up the Equation:**
- Substitute the value of \(\sin(30^\circ)\):
\[ \frac{R}{L} = \frac{1}{2} \]

6. **Find the Ratio:**
- Rearranging gives us:
\[ R = \frac{1}{2}L \]
- This can be expressed as:
\[ \frac{R}{L} = \frac{1}{2} \]
- Therefore, the ratio of radius to slant height is:
\[ R : L = 1 : 2 \]

### Final Answer:
The ratio of the radius to the slant height of the conical tent is \(1 : 2\).

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