Question

A lake on a map has an area of $0.15c{m^2}$. Work out the actual area of the lake where the scale is taken as $1cm = 20km$.

Answer 1

Hint: A map is a miniature representation of a geographical area. But there is a scale used in the map to keep the uniformity of the distances. Distance between any two points in the actual area can be found by multiplying the respective distance in the map by a fixed number. The number here is $20$. But in the case of an area of a place, we have to consider the square of this fixed number.

Useful formula:
For a given scale in a map,
$1cm = xkm \Rightarrow 1c{m^2} = {(xkm)^2} = {x^2}k{m^2}$

Complete step-by-step answer:
Given that the lake has an area $0.15c{m^2}$ on the map.
Also given the scale of the map as $1cm$ represents $20km$.
We are asked to find the actual area of the lake.
Scale of a map is the ratio of a distance on the map to the actual distance.
For a given scale in a map,
$1cm = xkm \Rightarrow 1c{m^2} = {(xkm)^2} = {x^2}k{m^2}$
So since the scale here is $1cm$ to $20km$, we get $0.15c{m^2}$ in map represents $0.15 \times {(20km)^2} = 0.15 \times 400k{m^2} = 60k{m^2}$ in actual.
So, the actual area which is represented by $0.15c{m^2}$ in the map is $60k{m^2}$.
$\therefore $ The answer is $60k{m^2}$.

Note: Here, one possible mistake a student can make is simply multiplying the given area by $20$ to get the actual area. But this is wrong. For finding the distance between two points in the map, we can do so. But in case of area, we have to consider the square as well.