A right circular cylinder of maximum volume is cut from a solid cube of edge $14\text{cm}$. The total surface of the cylinder is

When a right circular cylinder of maximum possible volume is carved out from a solid cube of side length $a=14\text{cm}$:The height ($h$) of the cylinder will be equal to the edge length of the cube:
$$h=a=14\text{cm}$$
The base diameter of the cylinder will also equal the edge length of the cube, meaning its radius ($r$) is half the side length:
$$r=\frac{a}{2}=\frac{14}{2}=7\text{cm}$$
The mathematical formula for the Total Surface Area ($\text{TSA}$) of a cylinder is:
$$\text{TSA}=2\pi r(r+h)$$
Using $\pi \approx \frac{22}{7}$, substitute the values into the formula:
$$\text{TSA}=2\times \frac{22}{7}\times 7\times (7+14)$$
$$\text{TSA}=2\times 22\times 21$$
$$\text{TSA}=44\times 21=924{\text{cm}}^2$$