Question

The radius of the earth is given $6.4\times {{10}^{6}}m$. What is the order of magnitude of the size of the earth?
A. ${{10}^{6}}m$
B. $6\times {{10}^{6}}m$
C. ${{10}^{7}}m$
D. $5\times {{10}^{6}}m$

Answer 1

As a first step you could recall what exactly the order of magnitude is. Then, you could make necessary changes in the given value of the radius depending on this definition of order of magnitude. By changes, we mean the conversion of the given value to the nearest power of ten.

Complete answer:
In the question we are given the value of radius of earth as $6.4\times {{10}^{6}}m$ and are asked to find the order of magnitude of the size of the earth.
Firstly we need to understand what exactly order of magnitude means. So, order of magnitude is basically an approximation of a value relative to some value that could be taken as a reference value, usually as powers of ten. Order of magnitude of a quantity could be defined as the smallest power of 10.
$N=a\times {{10}^{b}}$
Where, the value of $a$ lies somewhere between $\dfrac{1}{\sqrt{10}}$ and$\sqrt{10}$, then, $b$ will be the order of magnitude of N. The order of magnitude is always an integer.
Here, we are supposed to find the order of magnitude size of the earth with the help of the given value of its radius.
$R=6.4\times {{10}^{6}}m$
We could find the required order of magnitude from the given radius by rounding off the given value into the nearest power of ten. As per the above condition 6.4 cannot be taken as $a$. We see that 6.4 is more near to ${{10}^{1}}$ than ${{10}^{0}}$, so, the radius could be expressed as,
$R\approx {{10}^{1}}\times {{10}^{6}}\approx {{10}^{7}}m$
Therefore, we found the order of magnitude of the size of earth to be equal to ${{10}^{7}}m$.

Hence, option C is found to be the correct answer.

Note:
Order of magnitude of a quantity makes it possible for approximate comparisons. If the difference is just one order of magnitude, one quantity is said to be 10 times that of the other and when this difference is two, one is 100 times that of the other. But we should always remember that it is just an estimate of a variable.