If the radius of a sphere be douled, then the percentage increase in volume is

Options

A. `500%`
B. `700%`
C. `600%`
D. `800%`

Answer

To solve the problem of finding the percentage increase in the volume of a sphere when its radius is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the volume of a sphere**:
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.

2. **Calculate the initial volume**:
Let the initial radius of the sphere be \( r \). Therefore, the initial volume \( V \) is:
\[ V = \frac{4}{3} \pi r^3 \]

3. **Double the radius**:
If the radius is doubled, the new radius becomes \( 2r \).

4. **Calculate the new volume**:
Substitute the new radius into the volume formula:
\[ V' = \frac{4}{3} \pi (2r)^3 \]
Simplifying this gives:
\[ V' = \frac{4}{3} \pi (8r^3) = \frac{32}{3} \pi r^3 \]

5. **Relate the new volume to the initial volume**:
Notice that:
\[ V' = 8 \left(\frac{4}{3} \pi r^3\right) = 8V \]
This shows that the new volume is 8 times the initial volume.

6. **Calculate the increase in volume**:
The increase in volume is:
\[ \text{Increase} = V' - V = 8V - V = 7V \]

7. **Calculate the percentage increase in volume**:
The percentage increase is given by:
\[ \text{Percentage Increase} = \left(\frac{\text{Increase}}{\text{Initial Volume}}\right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left(\frac{7V}{V}\right) \times 100 = 7 \times 100 = 700\% \]

### Final Answer:
The percentage increase in the volume of the sphere when the radius is doubled is **700%**.