The total surface area of a solid hemisphere is $462c{{m}^{2}}$. Find its radius.

Hint: Use the formula for the total surface area of a solid hemisphere, that is $3\pi {{r}^{2}}$. Equate this with the given total surface area and solve the equation to find the radius.

Complete step-by-step answer:
A solid sphere has two surfaces, a curved surface and a flat surface. The total surface area of the hemisphere is the sum of the surface areas of both these surfaces. The curved surface area is given as $2{{\pi }^{2}}$ and the surface area of the flat surface at the bottom is given by the same formula as the area of a circle as it is circular in shape; that is, $\pi {{r}^{2}}$.

Thus, the total surface area of the solid hemisphere will be $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.

Equating this surface area with the total surface area given in the question, we get
$\begin{align}

  & 3\pi {{r}^{2}}=462c{{m}^{2}} \\

 & \Rightarrow \pi {{r}^{2}}=\dfrac{462}{3}c{{m}^{2}} \\

 & \Rightarrow \pi {{r}^{2}}=154c{{m}^{2}} \\

\end{align}$

Using the value of $\pi =\dfrac{22}{7}$ in this equation we get
$\begin{align}

  & \dfrac{22}{7}{{r}^{2}}=154c{{m}^{2}} \\

 & \Rightarrow {{r}^{2}}=154c{{m}^{2}}\times \dfrac{7}{22} \\

 & \Rightarrow {{r}^{2}}=49c{{m}^{2}} \\

\end{align}$

Solving the equation by taking the positive square root on both sides, we get $r=7cm$. Thus the radius of the given solid hemisphere is 7 cm.

Note: Since the hemisphere is solid, therefore the base area also needs to be considered and not only the curved surface. It is common to make this mistake of considering only the curved surface in calculation of the total surface area, and should be kept in mind while solving such questions.