The percentage increase in the surface area of a cube when each side is doubled is

Options

A. `50%`
B. `200%`
C. `150%`
D. `300%`

Answer

To find the percentage increase in the surface area of a cube when each side is doubled, we can follow these steps:

### Step 1: Calculate the original surface area of the cube.
The formula for the surface area \( S \) of a cube with side length \( a \) is given by:
\[ S = 6a^2 \]
Let the original side length of the cube be \( a \). Therefore, the original surface area is:
\[ S_{\text{original}} = 6a^2 \]

### Step 2: Calculate the new side length when it is doubled.
When each side of the cube is doubled, the new side length becomes:
\[ a_{\text{new}} = 2a \]

### Step 3: Calculate the new surface area of the cube.
Using the new side length, we can find the new surface area:
\[ S_{\text{new}} = 6(a_{\text{new}})^2 = 6(2a)^2 = 6 \cdot 4a^2 = 24a^2 \]

### Step 4: Calculate the increase in surface area.
To find the increase in surface area, we subtract the original surface area from the new surface area:
\[ \text{Increase in Surface Area} = S_{\text{new}} - S_{\text{original}} = 24a^2 - 6a^2 = 18a^2 \]

### Step 5: Calculate the percentage increase in surface area.
The percentage increase is calculated using the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Increase}}{\text{Original}} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{18a^2}{6a^2} \right) \times 100 = 3 \times 100 = 300\% \]

### Final Answer:
The percentage increase in the surface area of the cube when each side is doubled is **300%**.

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