If the radius a sphere becomes double, then find the percentage increase in the surface.

Answer

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

### Step 1: Write 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 initial surface area with the original radius.
Let the original radius be \( r \). Therefore, the initial surface area \( A_{\text{old}} \) is:
\[ A_{\text{old}} = 4\pi r^2 \]

### Step 3: Determine the new radius when it is doubled.
If the radius becomes double, the new radius \( r' \) is:
\[ r' = 2r \]

### Step 4: Calculate the new surface area with the new radius.
Now, substitute \( r' \) into the surface area formula:
\[ A_{\text{new}} = 4\pi (2r)^2 \]
Calculating this gives:
\[ A_{\text{new}} = 4\pi (4r^2) = 16\pi r^2 \]

### Step 5: Find the change in surface area.
Now, calculate the change in surface area:
\[ \text{Change in surface area} = A_{\text{new}} - A_{\text{old}} = 16\pi r^2 - 4\pi r^2 \]
This simplifies to:
\[ \text{Change in surface area} = (16 - 4)\pi r^2 = 12\pi r^2 \]

### Step 6: Calculate the percentage increase in surface area.
The percentage increase is given by the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Change in surface area}}{A_{\text{old}}} \right) \times 100 \]
Substituting the values we found:
\[ \text{Percentage Increase} = \left( \frac{12\pi r^2}{4\pi r^2} \right) \times 100 \]
The \( \pi r^2 \) cancels out:
\[ \text{Percentage Increase} = \left( \frac{12}{4} \right) \times 100 = 3 \times 100 = 300\% \]

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

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