The edge of a cube is increasing at a rate of 7 cm/s. The rate of change of area of the cube when its side is 3 cm is:

$252\, cm^2/s$

The correct answer is Option (4) → $252\, cm^2/s$

Let edge of cube be $x$ cm. Surface area of cube: $A = 6x^2$

Given $\frac{dx}{dt} = 7 \text{ cm/s}$

Rate of change of area: $\frac{dA}{dt} = \frac{d}{dt}(6x^2) = 12x \frac{dx}{dt}$

When $x = 3$ cm: $\frac{dA}{dt} = 12 \cdot 3 \cdot 7 = 252 \text{ cm}^2/\text{s}$