If each edge of a cube is increased by 50%, the percentage increase in its surface area is

Options

A. `150%`
B. `75%`
C. `100%`
D. `125%`

Answer

To solve the problem of finding the percentage increase in the surface area of a cube when each edge is increased by 50%, we can follow these steps:

### Step 1: Understand the formula for the surface area of a cube.
The surface area \( A \) of a cube with side length \( s \) is given by the formula:
\[ A = 6s^2 \]

### Step 2: Calculate the original surface area.
Let the original side length of the cube be \( s \). The original surface area \( A_1 \) is:
\[ A_1 = 6s^2 \]

### Step 3: Calculate the new side length after a 50% increase.
If each edge of the cube is increased by 50%, the new side length \( s' \) is:
\[ s' = s + 0.5s = 1.5s \]

### Step 4: Calculate the new surface area.
Now, we can calculate the new surface area \( A_2 \) using the new side length:
\[ A_2 = 6(s')^2 = 6(1.5s)^2 = 6 \times 2.25s^2 = 13.5s^2 \]

### Step 5: Find the increase in surface area.
The increase in surface area \( \Delta A \) is:
\[ \Delta A = A_2 - A_1 = 13.5s^2 - 6s^2 = 7.5s^2 \]

### Step 6: Calculate the percentage increase in surface area.
To find the percentage increase, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta A}{A_1} \right) \times 100 \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{7.5s^2}{6s^2} \right) \times 100 = \left( \frac{7.5}{6} \right) \times 100 = 125\% \]

### Conclusion:
The percentage increase in the surface area of the cube when each edge is increased by 50% is **125%**.

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