What will be the approximate change in the surface area of a cube of side xm caused by increasing the side by 2%

The correct option is 0.24x².

To find the approximate change in the surface area of a cube, we can use differentials. Let's break down the solution step-by-step:

Step 1: Formula for the Surface Area of a Cube
The surface area ($S$) of a cube with a side length of $x$ meters is given by the formula:

$S=6x^2$

Step 2: Differentiating the Surface Area with Respect to Side Length
To determine how the surface area changes with respect to the side length $x$, we find the derivative of $S$ with respect to $x$:

$\frac{dS}{dx}=\frac{d}{dx}\left(6x^2\right)=12x$

Step 3: Calculating the Change in Side Length
The side of the cube increases by $2\%$. The change in the side length, denoted as $dx$, is:

$dx=2\%\text{ of }x=0.02x$

Step 4: Finding the Approximate Change in Surface Area
Using differentials, the approximate change in surface area ($dS$) is given by:

$dS\approx \left(\frac{dS}{dx}\right)·dx$

Substituting the values we obtained in Step 2 and Step 3:

$dS\approx \left(12x\right)·\left(0.02x\right)$

$dS\approx 0.24x^2{\text{ m}}^2$

Therefore, the approximate change in the surface area of the cube is 0.24x² m².