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

Last updated date: 24th Mar 2023
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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.

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 $2\pi {{r}^{2}}+\pi {{r}^{2}}=3\pi {{r}^{2}}$.
\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.