The apprximate change in the volume of a cube of side x metres caused by increasing the side by `5%` is :

Options

A. `0.06x^(3)m^(3)`
B. `0.6x^(3)m^(3)`
C. `0.15 x^(3)m^(3)`
D. `0.9x^(3)m^(3)`.

Answer

To find the approximate change in the volume of a cube when the side length is increased 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 Change in Side Length
An increase of 5% in the side length \( x \) can be calculated as:
\[ \Delta x = 5\% \text{ of } x = \frac{5}{100} \times x = 0.05x \]

### Step 3: Use the Derivative to Find the Change in Volume
To find the change in volume \( \Delta V \), we can use the formula:
\[ \Delta V \approx \frac{dV}{dx} \cdot \Delta x \]
First, we need to find the derivative of the volume with respect to \( x \):
\[ \frac{dV}{dx} = 3x^2 \]

### Step 4: Substitute \( \Delta x \) into the Change in Volume Formula
Now, substituting \( \Delta x \) into the equation:
\[ \Delta V \approx 3x^2 \cdot (0.05x) \]

### Step 5: Simplify the Expression
Now, we can simplify the expression:
\[ \Delta V \approx 3x^2 \cdot 0.05x = 0.15x^3 \]

### Step 6: State the Final Answer
Thus, the approximate change in the volume of the cube when the side is increased by 5% is:
\[ \Delta V = 0.15x^3 \text{ cubic meters} \]