A map is drawn to a scale of $1:4000$.
- Find the length on map (in cm) that represents $5 \; km$ of ground length.
- Find the area on ground (in sq.m) that is represented by $4 \; cm^2$ on this map.
Scale of map $= 1 : 4000$
- i.e. $\;$ $1 \; cm$ on map $\equiv$ $4000 \; cm$ on ground
i.e. $\;$ $1 \; cm$ on map $\equiv$ $4000 \times 10^{-5} \; km$ on ground $\;\;$ $\left[\because \; 1 \; cm = 10^{-5} \; km\right]$
i.e. $\;$ $1 \; cm$ on map $\equiv$ $4 \times 10^{-2} \; km$ on ground
$\therefore \;$ $x \; cm$ on map $\equiv$ $5 \; km$ on ground
$\implies$ $x = \dfrac{5}{4 \times 10^{-2}} \; cm = 125 \; cm$
$\therefore \;$ $125 \; cm$ on map $\equiv$ $5 \; km$ on ground - $1 \; cm$ on map $\equiv$ $4000 = 4 \times 10^3 \; cm$ on ground
$\therefore \;$ $1 \; cm^2$ on map $\equiv$ $16 \times 10^6 \; cm^2$ on ground
$\therefore \;$ $4 \; cm^2$ on map $\equiv$ $p \; cm^2$ on ground
i.e. $\;$ $p = 4 \times 16 \times 10^6 = 64 \times 10^6 \; cm^2$ on ground
Now, $100 \; cm = 1 \; m$
i.e. $\;$ $10^4 \; cm^2 = 1 \; m^2$
$\therefore \;$ $64 \times 10^6 \; cm^2 = \dfrac{64 \times 10^6}{10^4} = 6400 \; m^2$
$\therefore \;$ $4$ sq.cm on map $\equiv$ $6400$ sq.m on ground