Question

In an Atlas a map occupies \[\dfrac{2}{5}\]th of a page with dimensions 25cm and 30cm respectively. If the real area of the map is 10800 sq. m the scale to which the map is drawn is
A. 1cm = 36m
B. 1cm = 26m
C. 1cm = 33m
D. 1cm = 6m

Answer 1

Hint:- Find the area of the map it acquires on the map using the data given in the question, Now this area represents the area of 10800 sq. m on the real map so compare them to find the area of 1c$m^2$ of the given divide that is divide these two, Now convert c$m^2$ into cm to get the desired answer.

Complete step-by-step answer:
As we know that the dimensions of the page of Atlas is 25cm and 30cm.
So, the area of the page of the Atlas will be = \[\left( {25 \times 30} \right)c{m^2} = 750c{m^2}\]
Now the map only occupies \[\dfrac{2}{5}\] of the page of the Atlas.
So, the area of the page occupied by the map will be = \[\dfrac{2}{5} \times 750c{m^2} = 300c{m^2}\]
Now the original area of the map is \[10800{m^2}\]
So, the area of \[300c{m^2}\] on the map represents \[10800{m^2}\] on the real map.
So, area of \[1c{m^2}\] on the map of Atlas represents \[\dfrac{{10800}}{{300}} = 36{m^2}\] of the real map.
Hence, \[36{m^2}\] of the original map is resented by \[1c{m^2}\] on the Atlas map.
But the scale is the measure of original map length represented on the Atlas map.
So, we had to take the root to find the scale for the map.
Scale = \[\sqrt {\dfrac{{36}}{1}} = 6m\]
Hence, 1cm on the map of Atlas represents 6m of the real map. So, scale will be 1cm = 6m.

Note:- Whenever we come up with this type of problem then there is no need to make the units of the area of the real map and the Atlas map the same. First, we had to find the area of the paper by multiplying the dimensions because the paper is rectangle and after that we had to find \[\dfrac{2}{5}\] of the page to get the area covered by map on Atlas and then we will divide that area by the real area of map to get how much area of the real map (in \[{m^2}\]) is shown by how much \[c{m^2}\] of the area of map in Atlas. Then we had to find the square root of that ratio because scale is the ratio of length shown by map of atlas to the original length not the ratio of areas. This will be the easiest and efficient way to find the solution of the problem.