Problem 90 - Applied Calculus

Question

Electronic commerce or e-commerce, buying and selling over the Internet, has been growing rapidly. The total value of U.S. e-commerce in recent years in billions of dollars is given by the exponential function \(f(x)=226(1.11)^{x}\), where \(x\) is the number of years since \(2012 .\) Predict total e-commerce in the year 2020 .

Step-by-Step Solution

Answer

The predicted e-commerce value in 2020 is approximately 484.44 billion dollars.

Step 1: Identify the function and the variable

We are given the exponential function \( f(x) = 226(1.11)^x \), which represents the total e-commerce value in billions of dollars. The variable \( x \) represents the number of years since 2012.

Step 2: Determine the value of x for 2020

Since \( x \) is the number of years since 2012, for the year 2020, \( x \) would be 2020 - 2012 = 8.

Step 3: Substitute x into the function

Substitute \( x = 8 \) into the function: \( f(8) = 226(1.11)^8 \).

Step 4: Calculate the expression

First, compute \((1.11)^8\). Then, multiply the result by 226: \[ f(8) = 226 imes (1.11)^8 \approx 226 imes 2.1436 \approx 484.44. \]

Step 5: Interpret the result

The result of \( f(8) \approx 484.44 \) indicates that the predicted total e-commerce value in the year 2020 is approximately 484.44 billion dollars.

Key Concepts

E-commerce Predictions

Predicting the future of e-commerce involves using mathematical models to foresee what the upcoming trends and values will be. The exponential function given in the exercise, \( f(x) = 226(1.11)^x \), helps make such predictions by showing the trend of growth over time. This function suggests that e-commerce is growing by 11% each year, a crucial insight when considering future business opportunities and challenges.

The year \(x\) measures the growth since 2012, so for 2020, \(x = 8\). This exemplifies that forecasting is not only about the present but also about how much growth has occurred since a baseline year. By plugging in \(x = 8\), we arrive at a prediction for 2020, guiding businesses to prepare and strategize efficiently for this swift growth.

• Exponential functions capture growth patterns, providing critical foresights.
• By understanding this trend, businesses can plan their e-commerce strategies better.

Calculating Future Values

Predicting e-commerce values requires calculating future values accurately using given functions. For this problem, the challenge is to calculate what the value of e-commerce will be in a year beyond the given baseline of 2012.

The given exponential model is implemented as \(f(x) = 226(1.11)^x\), which projects growth based on a consistent percentage increase. To find the future value for 2020, we determine \(x = 8\) (since 2020 is 8 years after 2012). After substituting \(x\) into the formula, the expression turns into \(f(8) = 226(1.11)^8\).

Next, compute \((1.11)^8\), which equals approximately 2.1436. Multiplying this by 226 gives the estimated value of approximately 484.44 billion dollars. Understanding this process helps with any problems requiring future value calculations using exponential growth:

• Recognize the base and growth rate for any exponential function.
• Apply these consistent growth models to other future predictions.

Interpreting Exponential Functions

Exponential functions are invaluable for interpreting growth patterns in various fields, including e-commerce. In our exercise, the function \(f(x) = 226(1.11)^x\) tells us how rapidly e-commerce grows over the years.

Here's what each part of the function represents:

• 226: The initial value of e-commerce in billions for the year 2012, serving as the starting point for calculations.
• 1.11: The growth factor, showing an 11% increase each year.
• \(x\): The variable that represents the number of years since 2012.

To interpret the calculated function for the year 2020, you substitute \(x = 8\), and the function shows 484.44 billion dollars, which highlights a strong and consistent growth pattern.

When you understand these components, you can identify how exponential functions provide a powerful tool to predict changes over time, not just in e-commerce, but in any scenario where growth occurs exponentially. Using functions like these, you can:

• Predict long-term growth effects.
• Plan strategic responses to changing economic conditions.