Question

A traditional unit of length in Japan is the ken \[\left( {1ken = 1.97m} \right)\]. What are the ratios of cubic kens to cubic meters? What is the volume of a cylindrical water tank of height $5.50kens$ and radius $3.00kens$ in cubic meters?

Answer 1

Hint: To solve this question we need to use the given relation to find out the required ratio. Then we have to calculate the volume in cubic kens and use the ratio obtained to get the result in cubic meters.

Formula Used $V = \pi {r^2}h$, where $V$ is the volume of a cylinder of $r$ radius and having $l$ length.

Complete step-by-step solution
It is given that
\[1ken = 1.97m\]
Taking cube both the sides, we get
$1ke{n^3} = 1.97m \times 1.97m \times 1.97m$
 \[1ke{n^3} = 7.645{m^3}\] (1)
Dividing on both the sides by $1{m^3}$, we get
 \[\dfrac{{ke{n^3}}}{{{m^3}}} = 7.645\]
Hence, the ratio of cubic kens to cubic meters is $7.645$
Now, we know that the volume of a cylinder is given by
$V = \pi {r^2}h$ (2)
According to the question, we have
$h = 5.50ken$
$r = 3ken$
Substituting these in (1), we get
$V = \pi {(3)^2}(5.5)$
$V = 98.96ke{n^3}$
Substituting from (1), we get the volume in cubic meters as
$V = 98.96 \times 7.645{m^3}$
On solving, we get
$V = 756.5{m^3}$

Hence, the volume in cubic meters is \[756.5{m^3}\]

Additional Information
The unit ken is a Japanese unit of length. It equals to six Japanese feet. It is commonly used as a measure in the architecture in Japan. It is used for measuring the intervals between the pillars of the buildings. In most of the Japanese houses, the surface areas of the floors are not measured in square meters. They are rather measured in the equivalent amounts of half of the square of ken. One half of a square ken is termed as one tatami.

Note
To make the calculations simple, we should not convert each dimension in the desired unit. Rather, we should first calculate the answer in the other units, and then use the multiplier to get the result in the desired unit. Doing so will save much time and energy.