Question

A cubical tank can hold 27000 litres of water. Find the dimension of its side.

Answer 1

Hint: First find out the amount space required to contain water of 27000 litre from the definition of 1 millilitre and use the formula for volume of a cube to what is the length of side to have that much amount of space.

Complete step by step answer:

 We know that a cuboid is a three dimensional object with six rectangular faces joined by 8 vertices. It has three different types of sides called length, breadth and height denoted $l,$$b$ and $h$. \[\]
A cube is cuboid with 6 square faces with all sides of equal length . It means $l=b=h=a$ (we denote).\[\]
The amount of space contained by a three dimensional object is measured by the quantity called volume. The amount of space that is occupied by a cuboid is the product of length, breadth and height. Mathematically, the volume of a cuboid is $l\times b\times h$. So we calculate the volume of a cube with denoting as $V$.
\[V=l\times b\times h=a\times a\times a={{a}^{3}}..(1)\]
The amount of liquid with unit density contained in the space by a cube of side of 1 cm is called 1 millilitre(ml). So the 1ml liquid is contained in $1cm\times 1cm\times 1cm=1c{{m}^{3}}$ of space. So now we find how much liquid is contained in a cube with the side 1metre(m).
 The amount of space contained by a cube of 1m is
\[V=1m\times 1m\times 1m=100cm\times 100cm\times 100cm=1000000c{{m}^{3}}\]
We know that 1000ml=1litre. $1c{{m}^{3}}$ of space contains 1 ml of liquid. So $1000000c{{m}^{3}}$ of space will contain liquid of \[\text{1}000000\text{ml}=\dfrac{\text{1}000000}{1000}\text{=1}000\text{ litre}\]
We find that a cube with side 1m or $1{{m}^{3}}$of space will contain 1000litre of liquid, in other words 1000litre of liquid will be contained by $1{{m}^{3}}$of space.
We know that at normal temperature and without any impurity the density of water is $1gm/c{{m}^{3}}$, that means unit density . \[\]
As given in the question, the tank of cubic shape contains 27000 litre of water. 1000litre of liquid will be contained by $1{{m}^{3}}$of space. The amount of space it requires to be contained 27000litre of water needs is $\dfrac{27000}{1000}=27{{m}^{3}}$
Let the side of the cubic tank be $a$ then its volume is ${{a}^{3}}$(from equation(1)). So
\[{{a}^{3}}=27\]
Let us take the cube root on both sides.
\[\begin{align}
  & {{\left( {{a}^{3}} \right)}^{\dfrac{1}{3}}}={{\left( 27 \right)}^{\dfrac{1}{3}}} \\
 & \Rightarrow a=3m \\
\end{align}\]

So the length of the side of tank is 3m.

Note: While solving the problem volume measurement of space and liquid we need to be careful of confusion between millilitre and centilitre. If we are asked to find the amount of water that will be flowed out when a body volume $v$ (in ${{m}^{3}}$) is submerged within the tank we can find it as $\left( V-v \right)1000$ litre. The question can also be framed to find the value of the total surface area of the cubic tank which for a cube is $6{{a}^{2}}$.