Question

Earth radius is $6371\,km$ . Show that the order magnitude of Earth’s radius is ${10^7}$

Answer 1

HintThe solution of given problem is based on following two points:
1) “Rounding off” the number
2) Theory of “order of magnitude”

 Complete step-by-step solution:First write the given number $\left[ {6371\,km} \right]$
Convert in SI system of units:
Since $1\,km = 1000\,m$
$\therefore 6371\,km = 6371 \times {10^3}m = 6371000\,m$
This number is multiplied and divided by ${10^6}$. This help to place decimal sign
\[
  6371000 \times \dfrac{{{{10}^6}}}{{{{10}^6}}}\,m \\
   = 6.371000 \times {10^6}\,m \\
 \]
Using rounding off rule
1) If the digit to be arranged is less than five then the proceeding digit is left unchanged. In the number the digits $1\& 0$ are less than $5$
So number can be written as $ = 6.37 \times {10^6}\,m$
Again the digit to be dropped is more than five then the proceeding digit is raised by one. Thus we write:
$6.37 \times {10^6}\,m \cong 6.4 \times {10^6}\,m$
The order of magnitude is the power of $\left[ {10} \right]$ required to represent the quantity i.e. $\left[ {x \times {{10}^n}} \right]$
Here $x$ is a number lies between $1\,\& \,10$
Since the last digit in given number is $\left[ {6.4} \right]$ which is greater than five i.e. $6.4 > 5$
Thus the power of $n$ is increased by $1$
In the given number
\[ \Rightarrow 6.4 \times {10^6} \simeq {10^7}\]
Hence the order of radius is $\left[ {{{10}^7}} \right]$

Note: such type of problem require more practice, that helping to learn the mathematical rules:
Remember:
1) Rounding off
2) Rule to find order of magnitude correctly