Question

In one fortnight of a given month, there was a rainfall of $$10cm$$ in a river valley. If the area of the valley is $$7280k{m^2}$$, show that the total rainfall was approximately equivalent to the addition of the normal water of three rivers each $$1072km$$ long, $$75m$$ wide and $$3m$$ deep.

Answer 1

Hint: Here we need to show that the volume of the river valley after the rainfall is approximately the same as the addition of water of three rivers of the same dimension. First we will calculate the volume of the river valley using the given details and then we will calculate the volume of water in each river using the given dimensions then we will compare both the volumes.

Complete step by step answer:
Given that the area of the river valley is 7280$k{m^2}$.
Length of the rainfall in the valley was 10$cm$.
If we can imagine the river valley as a rectangular region, the 10rainfall increases the height of the river valley bed by 10. Then the volume of rainfall is nothing but the volume of the three dimensional rectangular region with height 10and area 7280$k{m^2}$.
Since the unit of area is given in $km$ lets convert 10 into $km$.
We know that$1km = 100000cm$.
$$\eqalign{
  & \Rightarrow 1cm = \dfrac{1}{{100000}}km \cr
  & \Rightarrow 10cm = 10 * \dfrac{1}{{100000}}km = \dfrac{1}{{10000}}km \cr} $$
Volume of rainfall = area of the valley$ * $length of rainfall
$ \Rightarrow $Volume of rainfall $$ = 7280k{m^2} * \dfrac{1}{{10000}}km$$
$ \Rightarrow $ Volume of rainfall $$ = 0.728k{m^3}$$
Now we will calculate the volume of the river of dimension 1072$km$ long, 75$m$ wide and 3$m$ deep.
Again we need to convert the unit $m$into using the conversion,
$$\eqalign{
  & 1km = 1000m \cr
  & \Rightarrow 1m = \dfrac{1}{{1000}}km \cr
  & \cr} $$
Volume of river=length$ * $breadth$ * $height of the river
 Volume of river $$ = 1072 * (75 * \dfrac{1}{{1000}})*(3 * \dfrac{1}{{1000}})$$ $$k{m^3}$$
On simplification we get, Volume of one river = $$0.24k{m^3}$$
This gives the volume of one single river , in the question it is given that the water of three rivers are added therefore, Volume of 3 rivers can be obtained by multiplying the volume of the river by 3.
Volume of 3 rivers = $$3*0.24k{m^3}$$ $$ = 0.72k{m^3}$$
Since the calculated values of the volume of rainfall and volume of rivers is approximately the same we can conclude that the volume of rainfall was approximately equal to the volume of 3 rivers.

Note: In this kind of statement problems first list out all the given information and identify what has to be found. Since this kind of problem requires us to calculate certain values like area and volume, we must be familiar with the mensuration formulas of standard shapes. Make sure all the dimension units are given the same, if not convert the units to make all the units the same and then substitute them in the problem.