The edge of a cube is increasing at a rate of 7 cm/s. Find the rate of change of area of the cube when x=6 cm

The correct option is 504 cm²/s.

Let us denote the edge length of the cube by $x$ and its total surface area by $A$.
The total surface area of a cube with edge length $x$ is given by the formula:
$$A=6x^2$$

We are given that the edge of the cube is increasing at a rate of 7 cm/s. Therefore, the rate of change of the edge length with respect to time $t$ is:
$$\frac{dx}{dt}=7\text{cm/s}$$

To find the rate of change of the area $A$ with respect to time $t$, we differentiate both sides of the surface area equation with respect to $t$ using the chain rule:
$$\frac{dA}{dt}=\frac{d}{dt}\left(6x^2\right)$$
$$\frac{dA}{dt}=12x·\frac{dx}{dt}$$

Now, we substitute the given values when the edge length $x=6\text{cm}$ and $\frac{dx}{dt}=7\text{cm/s}$:
$$\frac{dA}{dt}=12·\left(6\right)·\left(7\right)$$
$$\frac{dA}{dt}=72·7=504{\text{cm}}^2\text{/s}$$

Thus, the rate of change of the area of the cube when the edge length is 6 cm is 504 cm²/s.