Question

The volume and total surface area of a solid hemisphere are equal in magnitude.

The volume is expressed in ${\mathrm{cm}}^3$ and the area is expressed in ${\mathrm{cm}}^2$.

Find the Radius of the Hemisphere.

Answer 1

Calculate the radius of the hemisphere.

Let the radius of the hemisphere be $r$.

The volume of the hemisphere is given by:

$V=\frac{2}{3}\mathrm{\pi }{r}^3$

The total surface area of the hemisphere is given by:

$A=3\mathrm{\pi }{r}^2$

Since the volume and total surface area of a solid hemisphere are equal in magnitude.

$$\begin{gathered} \therefore V=A \\ \Rightarrow \frac{2}{3}{\mathrm{\pi r}}^3=3{\mathrm{\pi r}}^2 \\ \Rightarrow 2{\mathrm{r}}^3-9{\mathrm{r}}^2=0 \\ \Rightarrow {\mathrm{r}}^2(2\mathrm{r}-9)=0 \\ \Rightarrow \mathrm{r}=\frac{9}{2}\mathrm{or}\mathrm{r}=0 \\ \Rightarrow \mathrm{r}=4.5\mathrm{cm}\mathrm{or}\mathrm{r}=0 \end{gathered}$$

Since the radius can not be zero.

Hence, the radius of the hemisphere is $4.5\mathrm{cm}$.