A traditional unit of length in Japan is the ken (${}^{1\mathrm{ken}=1.97\mathrm{m}}$). What are the ratios of (a) square kens to square meters and (b) cubic kens to cubic meters? What is the volume of a cylindrical water tank of height ${}^{5.50\mathrm{kens}}$and radius ${}^{3.00\mathrm{kens}}$in (c) cubic kens and (d) cubic meters?

$\mathbf{1}\mathbf{}\mathbf{ken}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{97}$

The height of the water tank is${}^{\mathbf{5}\mathbf{.}\mathbf{50}\mathbf{}\mathbf{kens}}$ .

The radius of the water tank is ${}^{\mathbf{3}\mathbf{.}\mathbf{30}\mathbf{}\mathbf{kens}}$.

In unit conversion, conversion factors are used to converts the units of measure for the same quantity.

The expression for the volume of a cylindrical water tank is,

$\mathit{V}\mathbf{=}\mathit{\pi }{\mathit{r}}^{\mathbf{2}}\mathit{h}$ … (i)

Here, ${}^{\mathbf{r}}$ is the radius and ${}^{\mathbf{h}}$is the height.

The ratio of square kens to square meters is,

Thus, the ratio of square kens to square meters is ${}^{\mathbf{3}\mathbf{.}\mathbf{88}\mathbf{}\mathbf{:}\mathbf{}\mathbf{1}}$.

The ratio of cubic kens to cubic meters is,

Thus, the ratio of cubic kens to cubic meters is${}^{\mathbf{7}\mathbf{.}\mathbf{65}\mathbf{}\mathbf{:}\mathbf{}\mathbf{1}}$

Using equation (i), the volume of water tank is calculated as:

Thus, the volume of the cylindrical water tank in cubic kens is${}^{\mathbf{156}\mathbf{}{\mathbf{kens}}^{\mathbf{3}}}$ .

Use the value of the volume of cylinder calculated in part (c) in cubic kens and covert it into cubic meters using the conversion factor.

Thus, the volume of cylindrical water tank in cubic meters is ${}^{\mathbf{1}\mathbf{.}\mathbf{19}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}\mathbf{}}{\mathbf{m}}^{\mathbf{3}}}$.