Find the approximate change in the volume of cube of side xm caused by increasing the side by 6%

The correct option is 0.18x³.

Let us understand how to find the approximate change in the volume of a cube using differentials.

First, let the side of the cube be denoted by x meters (m).
The volume V of a cube with side x is given by the formula:
$$V=x^3$$

The change in the side of the cube is an increase of 6%. Therefore, the change in the side, denoted by Δx (or dx), is:
$$dx=6\%\text{ of }x=\frac{6}{100}x=0.06x$$

The approximate change in the volume, denoted by dV, can be found using the differential:
$$dV=\frac{dV}{dx}\cdot dx$$

Differentiating the volume formula V = x³ with respect to x, we get:
$$\frac{dV}{dx}=3x^2$$

Now, substitute the values of $\frac{dV}{dx}$ and dx into the expression for dV:
$$dV=\left(3x^2\right)\cdot \left(0.06x\right)$$

Multiplying the terms:
$$dV=\left(3\cdot 0.06\right)x^3=0.18x^3$$

Thus, the approximate change in the volume of the cube is 0.18x³ cubic meters.