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

Options

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

Answer

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

### Step 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.

### Step 2: Calculate the original surface area.
Let the original radius of the sphere be \( R \). Then the original surface area \( A_1 \) is:
\[ A_1 = 4\pi R^2 \]

### Step 3: Calculate the new radius.
If the radius is doubled, the new radius \( R' \) will be:
\[ R' = 2R \]

### Step 4: Calculate the new surface area.
Now, we can find the new surface area \( A_2 \) using the new radius:
\[ A_2 = 4\pi (R')^2 = 4\pi (2R)^2 = 4\pi (4R^2) = 16\pi R^2 \]

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

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

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

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