The radius of the base and the height of a right circular cone are doubled. The volume of the cone will be

Options

A. 8 times of the previous volume
B. three times of the previous volume
C. `3sqrt(2)` times of the previous volume
D. 6 times of the previous volume

Answer

To solve the problem, we need to determine how the volume of a right circular cone changes when both its radius and height are doubled.

### Step-by-step Solution:

1. **Understand the formula for the volume of a cone**:
The volume \( V \) of a right circular cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height of the cone.

2. **Define the initial dimensions**:
Let the initial radius of the cone be \( r \) and the initial height be \( h \).

3. **Calculate the initial volume**:
Using the formula, the initial volume \( V_{\text{initial}} \) is:
\[ V_{\text{initial}} = \frac{1}{3} \pi r^2 h \]

4. **Double the radius and height**:
If the radius and height are both doubled, the new radius \( r' \) and new height \( h' \) will be:
\[ r' = 2r \quad \text{and} \quad h' = 2h \]

5. **Calculate the new volume**:
Substitute the new dimensions into the volume formula:
\[ V_{\text{final}} = \frac{1}{3} \pi (r')^2 (h') = \frac{1}{3} \pi (2r)^2 (2h) \]
Simplifying this gives:
\[ V_{\text{final}} = \frac{1}{3} \pi (4r^2)(2h) = \frac{1}{3} \pi (8r^2 h) \]

6. **Relate the new volume to the initial volume**:
We can express the new volume in terms of the initial volume:
\[ V_{\text{final}} = 8 \left(\frac{1}{3} \pi r^2 h\right) = 8 V_{\text{initial}} \]

7. **Conclusion**:
Therefore, the volume of the cone after doubling the radius and height is 8 times the initial volume.

### Final Answer:
The volume of the cone will be **8 times** the previous volume.

---