Q49P - Fundamentals Of Physics

Question

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?

Step-by-Step Solution

Answer

Step 1: Given data

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

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

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

Step 2: Understanding the conversion factor

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.

Step 3: (a) Determination of the ratio of square kens to square meters.

The ratio of square kens to square meters is, 

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

Step 4: (b) Determination of the ratio of cubic kens to cubic meters.

The ratio of cubic kens to cubic meters is,

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

Step 5: (c) Determination of the volume of cylindrical water tank in cubic kens

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

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

Step 6: (d) Determination of the volume of the cylindrical water tank in cubic meters

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 ${}^{\mathbf{1}\mathbf{.}\mathbf{19}\mathbf{\times }{\mathbf{10}}^{\mathbf{3}\mathbf{ }}{\mathbf{m}}^{\mathbf{3}}}$ .