Question

A solid cubical block of fine wood costs \[{\text{Rs}}.{\text{ 256}}\] at \[{\text{Rs}}.{\text{ 500}}\] per m cube. Find its volume and the length of each side.

Answer 1

Hint: A cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meetings at each vertex.
The cube is the only regular hexahedron and is one of the five Platonic solids. It has \[6\]faces, \[12\] edges, and \[8\] vertices.
Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains. Volume is often quantified numerically using the SI derived unit, the cubic meter.
\[{\text{Volume of the cube}} = {(side)^3}\]

\[{\text{Total Surface area }} = {\text{ 6 }}{\left( {{\text{ side}}} \right)^2}\]
Face diagonal \[ = \;\sqrt 2 a\]
Space diagonal = \[ = \;\sqrt 3 a\]
\[Cost = Volume({m^3}) \times Rate/{m^3}\]

Complete answer:

According to question
Cost of one m³ of cloth \[ = \] \[{\text{Rs}}.{\text{ 500}}\]
Cost of the cube \[ = \]\[{\text{Rs}}.{\text{ 256}}\]
Therefore,
On putting the value of Cost of one m³ of cloth = \[{\text{Rs}}.{\text{ 500}}\] and Cost of the cube =\[{\text{Rs}}.{\text{ 256}}\]in the formula

We get
Now to convert it into \[c{m^3}\] we multiply \[{\text{1}}000000\]to it.
Thus,
So, we have the volume.
Thus, the length of the side of the cube \[ = \sqrt[3]{{{\text{512}}000}} = 80cm\]

Note: The cube is also a square parallelepiped, an equilateral cuboid, and a right rhombohedron. It is a regular square prism in three orientations, and a trigonal, trapezohedron in four orientations.
The cube is dual to the octahedron. It has cubical or octahedral symmetry.
The cube is the only convex polyhedron whose faces are all squares.