Problem 11 - Precalculus

Question

The number of cell phones in China is exploding. In 2007 there were 487.4 million cell phone subscribers and the number is increasing at a rate of \(16.5 \%\) per year. How many cell phone subscribers are expected in \(2010 ?\) Use the formula \(N=N_{0} e^{r t},\) where \(N\) represents the number of cell phone subscribers. Let \(t=0\) correspond to 2007.

Step-by-Step Solution

Answer

Approximately 799.4 million cell phone subscribers are expected in 2010.

Step 1: Identify the Given Values

We are provided with the initial number of subscribers in 2007, which is \( N_0 = 487.4 \) million. The growth rate \( r \) is \( 16.5\% \) or \( 0.165 \) when expressed as a decimal. The time \( t \) between 2007 and 2010 is \( t = 3 \) years.

Step 2: Apply the Formula

We use the formula \( N = N_0 e^{r t} \) to find the future number of subscribers. Substitute the values: \( N = 487.4 \times e^{0.165 \times 3} \).

Step 3: Calculate the Exponential Growth

First, calculate the exponent: \( 0.165 \times 3 = 0.495 \). Then calculate \( e^{0.495} \). The value of \( e^{0.495} \) is approximately \( 1.641 \).

Step 4: Compute the Final Number of Subscribers

Substitute back to find \( N \): \( N = 487.4 \times 1.641 \). This gives \( N \approx 799.39 \).

Step 5: Round the Answer

Based on common practice in scenarios involving large numbers of people, we round the number of subscribers to the nearest tenth of a million, which results in about \( 799.4 \) million subscribers.

Key Concepts

Compound Interest Formula

In mathematics, the concept of compound interest is widely used to model exponential growth, such as the rapid proliferation of cell phone subscribers over time.
The compound interest formula is a powerful tool that can also describe how investments grow in value over time.
In the context of our exercise, the formula we use is:

• \(N = N_0 e^{rt}\)

Here:

• \(N\) is the future number of subscribers we want to find.
• \(N_0\) represents the initial number of subscribers, which is \(487.4\) million in 2007.
• \(r\) is the growth rate, expressed as a decimal. A rate of \(16.5\%\) becomes \(0.165\) when converted.
• \(t\) is time in years, with 2007 as \(t=0\).

Compound interest is all about exponential increase. This means things grow faster as time goes on because the growth applies to ever-larger amounts.
When you thoroughly understand how to manipulate this formula, you can predict how resources or investments might grow, given enough time and consistent rates.

Mathematical Modeling

Mathematical modeling is a method of representing real-world phenomena using mathematical concepts and formulas.
In the given problem, we are modeling the growth in the number of cell phone subscribers in China using an exponential function.
This model uses known data from 2007 and projects it forward based on a constant rate of growth.The task involves:

• Understanding the initial conditions — the number of subscribers, which is our starting point.
• Identifying the growth rate and expressing it in a usable format (as a decimal).
• Determining the time frame over which growth occurs (from 2007 to 2010, or 3 years).

When we apply mathematical modeling with these considerations, we can create a predictive formula, like the compound interest formula explained above.
This predicts that in 2010, there would be approximately \(799.4\) million subscribers, capturing the essence of how the rate affects the growth.

Growth Rate Calculation

The growth rate calculation is crucial in understanding and predicting how rapidly a quantity can increase over time.
Our focus is on how cell phone subscriptions grow in China, which we assume happens at a constant rate of \(16.5\%\) annually.
Let's break this down:

• The given growth rate is first converted for mathematical use: \(16.5\%\) becomes \(0.165\).
• Using the time (\(t\)) of 3 years, we calculated an exponent for growth: \(r \times t = 0.165 \times 3 = 0.495\).
• Applying exponential growth, we then computed \(e^{0.495}\), which equals approximately \(1.641\).

This calculation shows how much the initial number will increase by a factor derived from the growth rate and time.
Growth rates need constant monitoring and adjusting, as they can fluctuate due to real-world conditions like market saturation or technological advancements.