Question

A marketing team conducted a study on the use of smartphones. In a certain metropolitan area, there were 1.6 million smartphone users at the end of 2018. The marketing team predicted that the number of smartphone users would increase by 35 percent each year beginning in 2019 . If \(y\) represents the number of smartphone users in this metropolitan area after \(x\) years, then which of the following equations best models the number of smartphone users in this area over time? 1\. \(y = 1,600,000(1.35)^{x}\) 2\. \(y = 1,600,000(35)^{x}\) 3\. \(y= 35 x+1,600,000\) 4\. \(y= \)1.35 x+1,600,000$

Short answer

Option 1: y = 1,600,000 (1.35)^x

Step-by-step solution

- Understand the problem

Identify the initial number of smartphone users and the annual growth rate provided in the problem. The initial number of smartphone users at the end of 2018 is 1.6 million. The annual growth rate is 35 percent.

- Convert percentage to a decimal

Convert the annual growth rate from a percentage to a decimal. 35 percent is equivalent to 0.35 as a decimal.

- Identify the exponential growth formula

Recall that the formula for exponential growth is given by y = y_0 (1 + r)^x where y_0 is the initial amount (1.6 million), r is the growth rate (0.35), and x is the number of years.

- Substitute values into the exponential growth formula

Substitute the initial number of smartphone users (1,600,000), the growth rate (0.35), and the variable x representing the number of years into the exponential growth formula. This gives the model y = 1,600,000 (1 + 0.35)^x Simplify the expression inside the parentheses to y = 1,600,000 (1.35)^x

- Compare the equation with the given options

Compare the derived equation, y = 1,600,000 (1.35)^x, with the given choices. Option 1 is equivalent to this equation, which confirms that it is the correct model.

Key concepts

Growth Rate

Growth rate is a percentage measure of how much something increases over a particular period. In our example, the number of smartphone users increases by 35 percent each year. This rate tells us the speed at which the change happens. The growth rate also helps in predicting future values. It is crucial to identify this rate accurately when dealing with exponential growth problems to ensure correct calculations.

Exponential Function

An exponential function models situations where a quantity grows or decreases at a consistent percent rate per time period. The general form is given by \(y = y_0(1 + r)^x\). Here, \(y_0\) represents the initial amount, \(r\) is the growth rate, and \(x\) is the number of time periods. In our problem, the number of smartphone users follows an exponential growth pattern, so we use this formula to describe their increase over time. For example, substituting the initial number (1.6 million), growth rate (0.35), and time (years) into the formula gives us \(y = 1,600,000(1.35)^x\).

Mathematical Modeling

Mathematical modeling involves creating equations to represent real-world situations. The primary purpose is to make predictions and gain insights into the behavior of complex systems. In this exercise, we used the exponential growth formula to model the increase in smartphone users. By accurately substituting the given data into the formula, we can effectively predict future smartphone user numbers. The process includes:

• Identifying initial values and rates of change
• Choosing the appropriate mathematical formula
• Substituting values into the formula
• Simplifying the expression for the final model

Percentage to Decimal Conversion

In mathematics, percentages are often converted to decimals to facilitate calculations. To convert a percentage to a decimal, you divide by 100. For instance, 35 percent becomes 0.35. This conversion is essential in exponential growth calculations. Without it, the formulas would be incorrect, and the predictions inaccurate. Always ensure to convert percentage growth rates into decimal form before using them in your calculations. Substituting this conversion into our example, the annual growth rate of 35 percent properly converts to 0.35.