Find the approximate change in the volume V of a cube of side x metres caused by increasing the side by `5%`.

Answer

To find the approximate change in the volume \( V \) of a cube of side \( x \) meters caused by increasing the side by \( 5\% \), we can follow these steps:

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

### Step 2: Calculate the increase in side length
A \( 5\% \) increase in the side length \( x \) can be calculated as:
\[ \Delta x = 0.05x \]

### Step 3: Use the formula for approximate change in volume
The approximate change in volume \( \Delta V \) can be found using the derivative of the volume with respect to the side length:
\[ \Delta V \approx \frac{dV}{dx} \cdot \Delta x \]

### Step 4: Calculate the derivative of the volume
To find \( \frac{dV}{dx} \), we differentiate \( V = x^3 \):
\[ \frac{dV}{dx} = 3x^2 \]

### Step 5: Substitute \( \Delta x \) and \( \frac{dV}{dx} \) into the formula
Now, substitute \( \Delta x \) and \( \frac{dV}{dx} \) into the formula for \( \Delta V \):
\[ \Delta V \approx 3x^2 \cdot (0.05x) \]

### Step 6: Simplify the expression
Now, simplify the expression:
\[ \Delta V \approx 3x^2 \cdot 0.05x = 0.15x^3 \]

### Conclusion
Thus, the approximate change in the volume of the cube when the side is increased by \( 5\% \) is:
\[ \Delta V \approx 0.15x^3 \text{ cubic meters} \]

---