Question

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

Answer 1

Hint The 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