Problem 68 - Contemporary Precalculus

Question

The number of digital devices (such as MP3 players, handheld computers, cell phones, and PCs) in the world was approximately .94 billion in 1999 and is growing at a rate of \(28.3 \%\) a year. (a) Find the rule of a function that gives the number of digital devices (in billions) in year \(x,\) with \(x=0\) corresponding to 1999 (b) Approximately how many digital devices will be in use in \(2010 ?\) (c) If this model remains accurate, when will the number of digital devices reach 6 billion?

Step-by-Step Solution

Answer

Answer: The rule of the function is \(P(x) = 0.94(1 + 0.283)^x\). b) What is the estimated number of digital devices in use in 2010? Answer: There were approximately 3.251 billion digital devices in use in 2010. c) When will the number of digital devices in use reach 6 billion? Answer: The number of digital devices will reach 6 billion in approximately 2015.

Step 1: 1. Write down the exponential growth function model

An exponential growth model can be represented as: \(P(x) = P_0(1 + r)^x\) where \(P(x)\) represents the number of digital devices in year \(x\), \(P_0\) is the initial number of devices, \(r\) is the growth rate, and \(x\) is the number of years since 1999.

Step 2: 2. Replace initial number of devices and growth rate

The initial number of digital devices in 1999 is \(0.94\) billion, and the growth rate is \(28.3\%\). We should write the growth rate as a decimal, so \(r = 0.283\). Now we can replace \(P_0\) and \(r\) in the growth function: \(P(x) = 0.94(1 + 0.283)^x\)

Step 3: 3. Find the number of digital devices in 2010

To find the number of digital devices in use in 2010, we have to replace \(x\) by the number of years since 1999. So \(x = 2010 - 1999 = 11\). Now, we can plug in the value of \(x\) into the growth function: \(P(11) = 0.94(1 + 0.283)^{11}\) \(P(11) \approx 3.251\) billion devices So, there will be approximately \(3.251\) billion digital devices in use in 2010.

Step 4: 4. Find when the number of digital devices will reach 6 billion

Now, we want to find when the number of digital devices reaches 6 billion. We need to solve the equation: \(6 = 0.94(1 + 0.283)^x\) We'll first use logarithms to solve for x: \(\frac{6}{0.94} = (1 + 0.283)^x\) \(\log(\frac{6}{0.94}) = \log((1 + 0.283)^x)\) \(x\log(1 + 0.283) = \log(\frac{6}{0.94})\) Now, we can solve for \(x\): \(x = \frac{\log(\frac{6}{0.94})}{\log(1 + 0.283)}\) \(x \approx 15.6\) Since \(x\) represents the number of years since 1999, the number of digital devices will reach 6 billion in approximately \(15.6\) years after 1999, rounding up, in the year 2015.

Key Concepts

Exponential Function

Exponential functions are mathematical expressions used to model a variety of real-world phenomena, such as population growth or radioactive decay. An exponential function generally takes the form:

• \[ P(x) = P_0 (1 + r)^x \]

Here, \( P(x) \) represents the quantity at time \( x \) (in this example, the number of digital devices), \( P_0 \) is the initial quantity, and \( r \) is the growth rate expressed as a decimal. The variable \( x \) often indicates the number of years elapsed from a specific starting point, which in our exercise is the year 1999.
Exponential functions are characterized by their constant percentage growth or decay rates, meaning they grow by the same percentage over equal time intervals. This leads to rapid increases or decreases, making the function an ideal model for phenomena experiencing continuous growth.

Growth Rate

The term 'growth rate' refers to the rate at which a quantity, such as the number of digital devices, increases over time. In exponential growth models, the growth rate is a crucial component, determining how quickly the quantity expands.
In the context of our exercise, the growth rate is \(28.3\%\), which needs to be expressed as a decimal \( (r = 0.283) \) when inserted into our exponential function.
Key aspects of a growth rate include:

• The percentage change each period, illustrating how much the quantity has increased.
• Determining the steepness of the exponential curve, with higher growth rates resulting in a steeper curve.

Mathematically, the pronounced effect of compounding makes exponential growth models effective tools in predicting potential future outcomes based on historical data.

Logarithms

Logarithms are the mathematical inverse of exponentiation and serve as a powerful tool in solving equations involving powers. In the context of our exercise, logarithms are employed to ascertain the time it will take for a growing number of digital devices to reach a certain threshold.
The process includes:

• Rewriting the exponential equation.
• Using logarithmic identities and properties to isolate the variable \( x \), which represents time.
• Applying the characteristic that allows you to express an exponential term as \( x \log b = \, \log(y) \), where \( b \) is the growth factor.

In the worked example, logarithms help us determine the time required to reach 6 billion devices, showcasing their efficacy in tackling real-world exponential problems.

Mathematical Modeling

Mathematical modeling involves creating an abstract representation of a real-world process using mathematical concepts and language. In our exercise, we are modeling the growth of digital devices over time using an exponential function.
Some essential elements of mathematical modeling include:

• Defining variables that represent real-world quantities, such as the initial number of digital devices and growth rates.
• Building a formula or equation that encapsulates the dynamics of the situation.
• Using the model to make predictions about future events, like estimating the number of digital devices in a future year or determining when a certain quantity will be reached.

Effective mathematical modeling requires a thorough understanding of both the mathematical principles involved and the real-life scenario, ultimately enabling the translation of complex situations into solvable equations.