Chapter 7.2, Problem 43PPS

To determine

To find: the order of magnitude for the servers and active users. Compare the number of active users and servers.

Answer to Problem 43PPS

The order of magnitude for the servers is 5.

The order of magnitude for the active users is 7.

The number of active users is $1.53\times {10}^3$ times the number of servers.

Explanation of Solution

Given information:

Number of servers 750,000.

Number of active users is 1.15 billion.

Concept used:

Scientific notation:

A notation in which a given quantity can be expressed as a number with significant digits required for a specified degree of accuracy and multiplied by 10 to the appropriate power.

The power of 10 is called the order of magnitude.

Formula used:

Law of exponents:

  $\begin{array}{l}{a}^m\cdot a^n=a^{m+n}\\ \frac{a^m}{a^n}=a^{m-n}\end{array}$

Calculation:

Consider the number of servers.

750,000

So, the number of servers can be written as

  $\begin{array}{c}750,000=7.5\times 100000\\ =7.5\times {10}^5\end{array}$

Consider the number of active users.

1.15 billion

So, the number of servers can be written as

  $\begin{array}{c}1.15\text{ billion}=1150000000\\ =1.15\times 10000000\\ =1.15\times {10}^7\end{array}$

Now, compare the numbers for servers and active users as shown:

  $\begin{array}{c}\frac{1.15\times {10}^7}{7.5\times {10}^5}=1.53\times 10\times {10}^{7-5}\\ =1.53\times {10}^3\end{array}$