The radius of a sphere is doubled. The percentage of increase in its surface area is

Options

A. 75
B. 100
C. 300
D. 400

Answer

To find the percentage increase in the surface area of a sphere when its radius is doubled, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the formula for the surface area of a sphere**:
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
where \( r \) is the radius of the sphere.

2. **Calculate the initial surface area**:
Let the initial radius of the sphere be \( r \). The initial surface area \( A_1 \) is:
\[ A_1 = 4\pi r^2 \]

3. **Determine the new radius**:
If the radius is doubled, the new radius \( r' \) will be:
\[ r' = 2r \]

4. **Calculate the new surface area**:
Substitute the new radius into the surface area formula to find the new surface area \( A_2 \):
\[ A_2 = 4\pi (r')^2 = 4\pi (2r)^2 = 4\pi (4r^2) = 16\pi r^2 \]

5. **Find the increase in surface area**:
The increase in surface area \( \Delta A \) is given by:
\[ \Delta A = A_2 - A_1 = 16\pi r^2 - 4\pi r^2 = 12\pi r^2 \]

6. **Calculate the percentage increase**:
The percentage increase in surface area is calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A_1} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{12\pi r^2}{4\pi r^2} \right) \times 100 = \left( \frac{12}{4} \right) \times 100 = 3 \times 100 = 300\% \]

### Final Answer:
The percentage increase in the surface area of the sphere when the radius is doubled is **300%**.

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