For minimum curved surface area and given volume, the ration of the height and radius of base of a cone is :

Options

A. `1 : sqrt2`
B. `sqrt2 : 1`
C. `1 : 2`
D. None of these

Answer

To find the ratio of the height (H) and the radius (R) of a cone for minimum curved surface area given a fixed volume, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Variables**:
Let the radius of the base of the cone be \( R \), the height be \( H \), and the slant height be \( L \).

2. **Curved Surface Area Formula**:
The curved surface area \( S \) of a cone is given by:
\[ S = \pi R L \]

3. **Volume Formula**:
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi R^2 H \]

4. **Relate Slant Height to Radius and Height**:
From the Pythagorean theorem, we have:
\[ L^2 = R^2 + H^2 \implies L = \sqrt{R^2 + H^2} \]

5. **Substituting L in the Surface Area**:
Substitute \( L \) in the surface area formula:
\[ S = \pi R \sqrt{R^2 + H^2} \]

6. **Express H in terms of R and V**:
Rearranging the volume formula gives:
\[ H = \frac{3V}{\pi R^2} \]

7. **Substituting H in the Surface Area**:
Substitute \( H \) in the surface area formula:
\[ S = \pi R \sqrt{R^2 + \left(\frac{3V}{\pi R^2}\right)^2} \]
Simplifying gives:
\[ S = \pi R \sqrt{R^2 + \frac{9V^2}{\pi^2 R^4}} = \pi R \sqrt{\frac{\pi^2 R^6 + 9V^2}{\pi^2 R^4}} = \frac{\pi R}{\pi R^2} \sqrt{\pi^2 R^6 + 9V^2} \]

8. **Minimize the Surface Area**:
To minimize \( S \), we differentiate \( S \) with respect to \( R \) and set the derivative equal to zero:
\[ \frac{dS}{dR} = 0 \]

9. **Finding Critical Points**:
After differentiating and simplifying, we will find a relation between \( H \) and \( R \). This leads to:
\[ H^2 = 2R^2 \implies H = \sqrt{2} R \]

10. **Finding the Ratio**:
The ratio of height to radius is:
\[ \frac{H}{R} = \frac{\sqrt{2} R}{R} = \sqrt{2} \]

### Final Answer:
Thus, the ratio of the height to the radius of the cone for minimum curved surface area given a fixed volume is:
\[ \frac{H}{R} = \sqrt{2} : 1 \]