If the side of a cube is increased by 12%, by how much per cent does its volume increase?

Options

A. `40.4928%`
B. `50.524%`
C. `60.3292%`
D. `30.4928%`

Answer

To solve the problem of how much the volume of a cube increases when the side is increased by 12%, we can follow these steps:

### Step 1: Understand the volume of a cube
The volume \( V \) of a cube with side length \( a \) is given by the formula:
\[ V = a^3 \]

### Step 2: Calculate the new side length after the increase
If the side length is increased by 12%, the new side length \( a' \) can be calculated as:
\[ a' = a + 0.12a = 1.12a \]

### Step 3: Calculate the new volume with the increased side length
Now, we can find the new volume \( V' \) using the new side length:
\[ V' = (a')^3 = (1.12a)^3 \]
Using the property of exponents, we can simplify this:
\[ V' = 1.12^3 \cdot a^3 \]

### Step 4: Calculate \( 1.12^3 \)
Now we need to calculate \( 1.12^3 \):
\[ 1.12^3 = 1.404928 \]

### Step 5: Calculate the increase in volume
The increase in volume \( \Delta V \) can be calculated as:
\[ \Delta V = V' - V = 1.404928a^3 - a^3 = (1.404928 - 1)a^3 = 0.404928a^3 \]

### Step 6: Calculate the percentage increase in volume
To find the percentage increase in volume, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\Delta V}{V} \right) \times 100\% \]
Substituting the values we have:
\[ \text{Percentage Increase} = \left( \frac{0.404928a^3}{a^3} \right) \times 100\% = 0.404928 \times 100\% = 40.4928\% \]

### Final Answer
Thus, the volume of the cube increases by approximately **40.49%**.

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