Question

The order of magnitude of seconds in a day is \[{{10}^{x}}\] . What is x?

Answer 1

Hint: When a number is in the form of \[a\times {{10}^{n}}\], and \[10 > a >5\], then the order of magnitude is \[\left( n+1 \right)\]. Calculate the number seconds in a day. Then, express the results into the previously shown form. In this way, we can calculate the order of magnitude.

Complete step by step answer:
First, we need to calculate the number of seconds in a day.
Let’s do that in a step by step manner to avoid any calculation mistake,
1 minute = 60 seconds
1 hour = 60 minutes
 \[\begin{align}
  & \Rightarrow \text{1 hour}=60\times 60\text{ seconds} \\
 & \Rightarrow \text{1 hour}=3600\text{ seconds} \\
\end{align}\]
1 day = 24 hours
$\begin{align}
  & \Rightarrow \text{1 day = }24\times 3600\text{ seconds} \\
 & \Rightarrow \text{1 day = 864}00\text{ seconds} \\
\end{align}$
So, the number of seconds in a day is $\text{ 864}00\text{ seconds}$.
We can write the number as,
\[8.64\times {{10}^{4}}\]
We need to know a simple rule of thumb,
If a number is in the form \[a\times {{10}^{n}}\] where \[5 > a > 0\] , then the order of magnitude is n. If \[10 > a > 5\] , then the order of magnitude is \[\left( n+1 \right)\].

As we can see, the number of seconds in a day is,
\[8.64\times {{10}^{4}}\]
Now, compare with the number,
\[a\times {{10}^{n}}=8.64\times {{10}^{4}}\]
Hence, we can write as,
\[\begin{align}
  & a=8.64 \\
 & n=4 \\
\end{align}\]
As we can see, \[a\text{ }=\text{ }8.64\] and \[n\text{ }=\text{ }4\] .
So, we can say that \[10 > a > 5\] . As a result, the order of magnitude is 5.
So, we can modify the number in the following way,
\[0.864\times {{10}^{5}}\]
Hence, the order of magnitude is 5 and the value of x is 5 as well.

Note:
If a number is in the following form \[a\times {{10}^{n}}\],
Where, \[a\] is called the mantissa,
              \[10\] is the base,
       and \[n\] is the power.
You should be accustomed to the scientific methodologies. You should not consider the order of magnitude you can see in the final result. You need to make sure that the mantissa of the value is in between 5 and 10. Otherwise, the order of magnitude should be one more than the value in the final product.