A map is drawn to a scale of $1 : 20000$
- The distance between the two towns on the map is $9 \; cm $. Calculate the actual distance between the towns in km.
- Calculate the actual area in $m^2$ which represents $8 \; cm^2 $ on the map.
- Scale of map is $\;$ $1 : 20000$
i.e. $\;$ $1 \; cm$ on the map corresponds to an actual distance of $\;$ $20000 \; cm$
$\therefore \;$ $9 \; cm$ on the map corresponds to an actual distance of $\;$ $\dfrac{9 \times 20000}{1} = 180000 = 1.8 \times 10^5 \; cm$
Now, $\;$ $10^5 \; cm = 1 \; km$
$\therefore \;$ $1.8 \times 10^5 \; cm = 1.8 \; km$
$\therefore \;$ Actual distance between the towns $= 1.8 \; km$ - Scale of map is $\;$ $1 : 20000$
i.e. $\;$ $1 \; cm$ on the map corresponds to an actual distance of $\;$ $20000 = 2 \times 10^4 \; cm$
i.e. $\;$ $1 \; cm^2$ on the map corresponds to an actual area of $\;$ $\left(2 \times 10^4\right)^2 = 4 \times 10^8 \; cm^2$
$\therefore \;$ $8 \; cm^2$ on the map corresponds to an actual area of $\;$ $\dfrac{8 \times 4 \times 10^8}{1} = 3.2 \times 10^9 \; cm^2$
Now, $\;$ $10^4 \; cm^2 = 1 \; m^2$
$\therefore \;$ $3.2 \times 10^9 \; cm^2 = \dfrac{3.2 \times 10^9 \times 1}{10^4} = 3.2 \times 10^5 \; m^2$
$\therefore \; $ Actual area $= 3.2 \times 10^5 \; m^2$