The radius of base and the volume of a right circular cone are doubled. The ratio of the length of the larger cone to that of the smaller cone is :

Options

A. `1:4`
B. `1:2`
C. `2:1`
D. `4:1`

Answer

To solve the problem, we need to find the ratio of the lengths (heights) of two cones: the smaller cone and the larger cone, given that the radius of the base and the volume of the cone are both doubled.

### Step-by-Step Solution:

1. **Define the dimensions of the smaller cone:**
- Let the radius of the base of the smaller cone be \( r \).
- Let the height of the smaller cone be \( h \).

2. **Calculate the volume of the smaller cone:**
- The formula for the volume \( V \) of a right circular cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
- Therefore, the volume of the smaller cone is:
\[ V_{\text{small}} = \frac{1}{3} \pi r^2 h \]

3. **Determine the dimensions of the larger cone:**
- According to the problem, the radius of the base of the larger cone is doubled:
\[ \text{Radius of larger cone} = 2r \]
- The volume of the larger cone is also doubled:
\[ V_{\text{large}} = 2 \times V_{\text{small}} = 2 \left( \frac{1}{3} \pi r^2 h \right) = \frac{2}{3} \pi r^2 h \]

4. **Calculate the volume of the larger cone using its dimensions:**
- The volume of the larger cone can also be expressed as:
\[ V_{\text{large}} = \frac{1}{3} \pi (2r)^2 H \]
- Simplifying this gives:
\[ V_{\text{large}} = \frac{1}{3} \pi (4r^2) H = \frac{4}{3} \pi r^2 H \]

5. **Set the two expressions for the volume of the larger cone equal to each other:**
- From the previous steps, we have:
\[ \frac{2}{3} \pi r^2 h = \frac{4}{3} \pi r^2 H \]

6. **Cancel out common terms:**
- We can cancel \( \frac{1}{3} \pi r^2 \) from both sides (assuming \( r \neq 0 \)):
\[ 2h = 4H \]

7. **Solve for the ratio of heights:**
- Rearranging gives:
\[ H = \frac{2h}{4} = \frac{h}{2} \]
- Therefore, the ratio of the height of the larger cone \( H \) to the height of the smaller cone \( h \) is:
\[ \frac{H}{h} = \frac{h/2}{h} = \frac{1}{2} \]

8. **Final ratio:**
- The ratio of the length (height) of the larger cone to that of the smaller cone is:
\[ \text{Ratio} = \frac{H}{h} = 2:1 \]

### Conclusion:
The ratio of the length of the larger cone to that of the smaller cone is \( 2:1 \).