Question

A unit of the area often used in measuring land areas is the hectare, defined as ${10^4}{m^2}$. An open-pit coal mine consumes 75 hectare of land, down to a depth of 26 m, each year. What volume of earth, in cubic kilometers, is removed in this time?

Answer 1

Hint: We need to express the area of the coal pit in metres square. The volume is given by the product of the area and the depth of the mine.

Formula used: The formulae used in the solution are given here.
The volume is given by, $V = A \times d$ where $A$ is the area and $d$ is the depth of the open-pit coal mine.

Complete step by step answer:
The amount of space, measured in cubic units, that an object or substance occupies is called volume. Two-dimensional doesn’t have volume but has area only. It has been given that, an open-pit coal mine consumes 75 hectare of land, down to a depth of 26 m, each year. It has also been given that one hectare is defined as ${10^4}{m^2}$.
Mathematically, $1hectare = {10^4}{m^2}$.
The area of the open-pit coal mine is 75 hectares.
Thus, mathematically we can see that $75hectares = 75 \times {10^4}{m^2}$.
The depth of the open-pit coal mine is 26m.
The volume is given by the product of the area and the depth of the mine. The volume of earth, is removed in this time is given by, $V = A \times d$ where $A$ is the area and $d$ is the depth of the open-pit coal mine.
Now as, $A = 75 \times {10^4}{m^2}$ and $d = 26m$, we substitute these values into the equation above.
$\therefore V = 75 \times {10^4} \times 26$
$ \Rightarrow V = 1950 \times {10^4}{m^3}$
By the laws of the metric system, one kilometre is equivalent to one thousand metres.
Mathematically, $1km = {10^3}m$.
Thus, $1m = {10^{ - 3}}km.$
For area in square metres,
$1{m^2} = {\left( {{{10}^{ - 3}}km} \right)^2} = {10^{ - 9}}k{m^2}$.
To express the volume in terms of cubic kilometres, we multiply the result in metre square by ${10^{ - 9}}$. Thus the volume of earth, in cubic kilometres removed is $V = 1950 \times {10^4} \times {10^{ - 9}}k{m^3}$.

Note: There are 100 hectares in one square kilometre. An acre is about $0.405$ hectare and one hectare contains about $2.47$ acres.