Problem 32 - C++ How to Program

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)?

Step-by-Step Solution

Answer

It will take approximately 15 months for Facebook to grow its user base to one billion users and approximately 29 months to reach two billion users.

Step 1: Understand the growth model

Facebook's user base grows by 5% every month. To model this compound growth, use the formula for exponential growth: \(A = P(1 + r)^t\), where \(A\) is the amount after growth, \(P\) is the initial amount, \(r\) is the growth rate per time period, and \(t\) is the number of time periods.

Step 2: Set up the equation for one billion users

First, we'll set up the equation for reaching one billion users. With \(P = 500\) million, \(A = 1000\) million (one billion), and \(r = 0.05\) (5% monthly growth), we have \(1000 = 500(1 + 0.05)^t\).

Step 3: Solve for the number of months to reach one billion

Solve the equation: \(2 = (1.05)^t\). Take the logarithm of both sides to get \(\log(2) = t \cdot \log(1.05)\). Then isolate \(t\): \(t = \frac{\log(2)}{\log(1.05)}\).

Step 4: Calculate the number of months for one billion users

Using a calculator, \(t ≈ \frac{0.3010}{0.0212} ≈ 14.2\) months. Since we cannot have a fraction of a month, we round up to 15 months for the user base to double.

Step 5: Set up the equation for two billion users

Next, setup the equation to reach two billion users. Now \(A = 2000\) million (two billion), keeping everything else the same: \(2000 = 500(1 + 0.05)^t\).

Step 6: Solve for the number of months to reach two billion

Solve the equation: \(4 = (1.05)^t\). Take the logarithm of both sides to get \(\log(4) = t \cdot \log(1.05)\). Then isolate \(t\): \(t = \frac{\log(4)}{\log(1.05)}\).

Step 7: Calculate the number of months for two billion users

Using a calculator, \(t ≈ \frac{0.6021}{0.0212} ≈ 28.4\) months. Since we cannot have a fraction of a month, we round up to 29 months for the user base to quadruple.

Key Concepts

Exponential Growth Formula

Understanding the exponential growth formula is crucial when analyzing scenarios where quantities increase at a consistent rate over time, such as population growth, inflation, and compound interest. In our Facebook user base growth example, the formula is expressed as A = P(1 + r)^t.

In this equation, A represents the final amount after growth, P stands for the initial amount, r signifies the growth rate per period, and t is the number of time periods. To apply this to the Facebook scenario, we substitute our values accordingly, which enables us to forecast the number of users after a specific time frame given the steady growth rate.

Logarithmic Equations

When the exponential growth equation needs to be solved for the time variable (t), we often resort to logarithms. Logarithmic equations are the inverse operations of exponentiation and are pivotal in unraveling the duration it takes to reach a certain level of growth.

For instance, in our problem, we manipulated the equation 2 = (1.05)^t by taking the logarithm on both sides, which allowed us to isolate t and find out how long it would take for the Facebook user base to hit one billion. This process simplifies the computation and is an essential tool in various scientific and engineering disciplines where exponential relationships are common.

Growth Rate Calculation

Growth rate calculation is essential for projecting future values based on current data. By knowing the percentage increase per time unit, businesses and investors can make informed decisions. In our Facebook example, we are given a monthly growth rate of 5%, represented as 0.05 in our calculation.

Using the formula for exponential growth, we can quantify future outcomes, such as the time required to double or quadruple the initial user base. Understanding and accurately calculating growth rates facilitate strategic planning and business forecasting.

C++ Programming Concepts

Grasping C++ programming concepts can empower students to model and solve complex problems computationally, including those involving exponential growth. To tackle problems like the Facebook user base growth, students can implement loops, functions, and other control structures in C++.

For instance, one could create a function that accepts the initial user count, growth rate, and target user count, subsequently employing a loop to iterate through the months while the user count multiplies accordingly. This demonstrates how programming can provide practical solutions to theoretical exercises and is an invaluable skill in the realm of computer science and data analysis.