A map of a square plot of land is drawn to a scale of $1 : 2500$. If the area of the plot on the map is $72 \; cm^2$, find
- the actual area of the land in square kilometer;
- the length of the diagonal in the actual plot of land in meters.
-
Scale of square plot of land $= 1 : 2500$
On Map Actual $1 \; cm$ $2500 \; cm$ $1 \; cm^2$ $625 \times 10^4 \; cm^2$ $72 \; cm^2$ $x \; cm^2$
$\therefore \;$ $x = 72 \times 625 \times 10^4 = 45000 \times 10^4 = 45 \times 10^7 \; cm^2$
i.e. $\;$ Actual area of the land $= 45 \times 10^7 \; cm^2$
Now, $\;$ $1000 \; m \equiv 1 \; km$
$\therefore \;$ $10^6 \; m^2 \equiv 1 \; km^2$
$\left[\text{Note: } 1 \; m \equiv 100 \; cm \implies 1 \; m^2 = 10^4 \; cm^2 \right]$
i.e. $\;$ $10^6 \times 10^4 \; cm^2 \equiv 1 \; km^2$
i.e. $\;$ $10^{10} \; cm^2 \equiv 1 \; km^2$
$\therefore \;$ $45 \times 10^7 \; cm^2 \equiv \dfrac{45 \times 10^7}{10^{10}} = 45 \times 10^{-3} = 0.045 \; km^2$
$\therefore \;$ Actual area of the land in square kilometer $= 0.045 \; km^2$ -
Let the length of the side of the plot on the map be $= s \; cm$
Then, area of the square plot on the map $= s^2 \; cm^2$
Given: Area of the plot on the map $= 72 \; cm^2$
i.e. $\;$ $s^2 = 72$ $\implies$ $s = \sqrt{72} \; cm$
$\therefore \;$ Length of the diagonal of the square plot on the map
$= \ell = \sqrt{\left(\sqrt{72}\right)^2 + \left(\sqrt{72}\right)^2} = \sqrt{72 + 72} = \sqrt{144} = 12 \; cm$
On Map Actual $1 \; cm$ $2500 \; cm$ $12 \; cm$ $x \; cm$
$\therefore \;$ $x = 12 \times 2500 = 30000 = 3 \times 10^4 \; cm$
i.e. $\;$ length of the diagonal in the actual plot of land $= 3 \times 10^4 \; cm$
Now, $\;$ $100 \; cm \equiv 1 \; m$
$\therefore \;$ $3 \times 10^4 \; cm \equiv \dfrac{3 \times 10^4}{10^2} = 3 \times 10^2 = 300 \; m$
$\therefore \;$ length of the diagonal (in meter) in the actual plot of land $= 300 \; m$