Question

The volume of a hemisphere varies directly as the cube of its radius. Volume of a hemisphere whose radius is 1 cm. is $\frac{44}{21}{\mathrm{cm}}^3$. Find the volume of the hemisphere having radius as 7 cm.

Answer 1

$$\begin{gathered} \mathrm{Let}V\mathrm{and}r\mathrm{denote}\mathrm{the}\mathrm{volume}\mathrm{and}\mathrm{radius}\mathrm{of}\mathrm{the}\mathrm{hemisphere},\mathrm{respectively}. \\ Thus,wehave: \\ V\alpha r^3 \\ \therefore V=k{r}^3\left(k\mathrm{is}\mathrm{constant}.\right) \\ \mathrm{Also}, \\ \frac{44}{21}=k\times 1^3\left(V=\frac{44}{21}{\mathrm{cm}}^{3}andr=1\mathrm{cm}\right) \\ \Rightarrow k=\frac{44}{21} \\ \therefore V=\frac{44}{21}{r}^3\left(\mathrm{Equation}\mathrm{of}\mathrm{variation}\right) \\ \Rightarrow V=\frac{44}{21}\times 7^3\left(\because r=7\mathrm{cm}\right) \\ \Rightarrow V=718.66{\mathrm{cm}}^3 \end{gathered}$$