The approximate change in the volume of a cube of side x metres caused by increasing the side by `3%` is :

Options

A. `0.06x^(3)m^(3)`
B. `0.6x^(3)m^(3)`
C. `0.09x^(3)m^(3)`
D. `0.9x^(3)x^(3)`

Answer

To find the approximate change in the volume of a cube when the side length is increased by 3%, we can follow these steps:

### Step 1: Understand the problem
We have a cube with a side length of \( x \) meters. We need to find the change in volume when the side length is increased by 3%.

### Step 2: Write down the formula for the volume of a cube
The volume \( V \) of a cube with side length \( x \) is given by:
\[ V = x^3 \]

### Step 3: Calculate the change in side length
An increase of 3% in the side length \( x \) can be calculated as:
\[ \Delta x = 0.03x \]

### Step 4: Find the derivative of the volume with respect to the side length
To find the approximate change in volume, we need to calculate the derivative of the volume with respect to \( x \):
\[ \frac{dV}{dx} = \frac{d}{dx}(x^3) = 3x^2 \]

### Step 5: Use the derivative to find the approximate change in volume
The approximate change in volume \( \Delta V \) can be expressed as:
\[ \Delta V \approx \frac{dV}{dx} \cdot \Delta x \]
Substituting the values we have:
\[ \Delta V \approx 3x^2 \cdot (0.03x) \]

### Step 6: Simplify the expression
Now, we simplify the expression:
\[ \Delta V \approx 3x^2 \cdot 0.03x = 0.09x^3 \]

### Conclusion
The approximate change in the volume of the cube when the side is increased by 3% is:
\[ \Delta V \approx 0.09x^3 \]