Question

(Facebook User Base Growth) According to CNNMoney.com, Facebook hit 500 million users in July of 2010 and its user base has been growing at a rate of \(5 \%\) per month. Using the compound-growth technique you learned in Fig. 5.6 and assuming this growth rate continues, how many months will it take for Facebook to grow its user base to one billion users? How many months will it take for Facebook to grow its user base to two billion users (which, at the time of this writing, was the total number of people on the Internet)?

Short answer

It will take approximately 14.21 months for Facebook to grow its user base to one billion users and approximately 28.41 months to grow to two billion users.

Step-by-step solution

Identify the Compound Growth Formula

Since we are dealing with compound growth at a constant rate, we can use the formula for compound growth which is given by: \[ A = P \times (1 + r)^n \]where:- \( A \) is the amount of users after \( n \) periods,- \( P \) is the principal amount (initial user base),- \( r \) is the monthly growth rate as a decimal,- \( n \) is the number of periods (months in this case).

Solve for the Number of Months to Reach One Billion Users

We need to find the number of months \( n \) it will take for the user base to grow from 500 million to 1 billion (\( A = 1000 \times 10^6 \) users). Use the formula:\[ 1000 \times 10^6 = 500 \times 10^6 \times (1 + 0.05)^n \]Solve for \( n \) using logarithms:\[ \frac{1000 \times 10^6}{500 \times 10^6} = (1 + 0.05)^n \]\[ 2 = 1.05^n \]\[ n = \frac{\log(2)}{\log(1.05)} \]

Calculate the Exact Number of Months to Double

Using a calculator, find the value of \( n \) as follows:\[ n = \frac{\log(2)}{\log(1.05)} \approx 14.2067 \]The exact number of months needed to double the user base to one billion is approximately 14.21 months.

Solve for the Number of Months to Reach Two Billion Users

Using the same logic, now solve for the number of months to reach two billion users:\[ 2000 \times 10^6 = 500 \times 10^6 \times (1 + 0.05)^n \]Simplifying gives us\[ 4 = 1.05^n \]and solving for \( n \) gives\[ n = \frac{\log(4)}{\log(1.05)} \]

Calculate the Exact Number of Months to Quadruple

Calculate the value of \( n \) to find out the months needed to reach two billion users:\[ n = \frac{\log(4)}{\log(1.05)} \approx 28.4133 \]The exact number of months needed to quadruple the user base to two billion is approximately 28.41 months.

Key concepts

Mathematical Modeling

Mathematical modeling is a process that utilizes mathematical language to describe real-world scenarios. In the case of the Facebook user base growth problem, mathematical modeling is employed to predict the future growth of the company's users given current trends.

Through constructing a model based on the initial conditions—the starting point of 500 million users—and the defined growth rate of 5% per month, we can simulate the situation and make predictions. This model becomes a powerful tool that helps us answer questions about the future without the need for trial and error or waiting for time to pass in real life.

Understanding how to set up a model is crucial. You begin by defining variables that represent core aspects of the problem: 'P' for initial user base (principal amount), 'r' for monthly growth rate, and 'A' for the predicted future user base. This sets the stage for applying mathematical formulas to forecast outcomes, relying on mathematical theory to provide reliable predictions that inform business decisions or help understand trends.

Exponential Function

The exponential function is central to understanding compound growth, as it describes processes that increase at a rate proportional to the current value. For example, a user base that grows by a constant percentage over time is accurately modeled with an exponential function of the form \( A = P \times (1 + r)^n \)

In our Facebook case, \( P \), the initial user base, is a known value – 500 million users. The growth rate 'r' is 5%, which when converted to decimal for mathematical operations is 0.05. The variable 'n' represents time, which in this scenario is measured in months. Crucially, the exponential growth doesn't just add the same number of users each month; it adds an increasing number, because each month's growth is 5% of the ever-larger preceding total.

The exponential function reflects this by raising the growth rate to the power of 'n', the number of periods. This represents the compound nature of the growth, where each period's growth builds upon the previous period's total.

Logarithmic Calculation

Logarithmic calculation is a key technique in solving problems involving exponential growth, as it allows us to solve for the exponent—that is, the time period 'n' in our Facebook user growth problem. When we have \( A = P \times (1 + r)^n \) and we want to find 'n', logarithms help us to 'undo' the exponentiation.

To solve \( 2 = 1.05^n \) for 'n', we take the logarithm of both sides. Because logarithms tell us the power to which a number must be raised to produce another number, we can rearrange the equation to solve for 'n' as \( n = \frac{\log(2)}{\log(1.05)} \).

Using logarithmic calculation in this context allows us to answer the question of time required to reach a certain number of users. By utilizing a calculator with logarithmic functions, solving such time-based problems is made practical, bypassing the need for iterative or intuitive methods which might not be precise or feasible.