An electric device delivers a current of [tex]$15.0 A$[/tex] for 30 seconds. How many electrons flow through it?
Answer (1)
The amount owed decreases by approximately the same amount every 6 months.
A linear model predicts that the amount owed will eventually be negative.
An exponential model predicts that the amount owed will never be 0.
The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative. The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
Explanation
Understanding the Problem We are given a table of data showing the amount owed at different months and asked to determine which statement is true regarding the fit of linear and exponential models to the data.
Analyzing the Differences First, let's examine the differences in the amount owed between consecutive months. The amount owed decreases by approximately the following amounts every 6 months: 2700 − 2110 = 590 , 2110 − 1500 = 610 , 1500 − 850 = 650 , 850 − 220 = 630 . These differences are relatively close, suggesting a linear model might be a reasonable fit.
Considering a Linear Model Now, let's consider what a linear model would predict. If the amount owed continues to decrease at roughly the same rate, at some point the amount owed will become negative. We can estimate when this occurs by fitting a linear regression to the data. Using the data points (6, 2700), (12, 2110), (18, 1500), (24, 850), and (30, 220), we find the linear regression line. The x-intercept of the linear regression line is approximately 32.24. This means that according to the linear model, the repayment amount will eventually be negative (around month 32).
Considering an Exponential Model Next, let's consider an exponential model. An exponential model has the form y = A e b x , where y is the amount owed, x is the number of months, and A and b are constants. Using the data, we can find the exponential regression. The exponential regression is approximately y = 6494.41 e − 0.0987 x . In an exponential model, the amount owed will approach 0 as time goes on, but it will never actually reach 0. However, the question states that the amount owed will never be 0, which is true for an exponential model.
Conclusion Comparing the models, the key difference is that the linear model predicts a negative amount owed eventually, while the exponential model predicts the amount owed will never be 0. Since the amount owed cannot be negative, the exponential model is a better representation of the situation in the long run. Therefore, the statement 'The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative' is true.
Final Answer Therefore, the correct answer is: The exponential model better represents the situation because according to the linear model, the repayment amount will eventually be negative.
Examples
Regression models are used in various real-life scenarios, such as predicting sales, analyzing trends, and making forecasts. For instance, a company might use regression analysis to predict future sales based on historical data, taking into account factors like advertising spend and seasonal variations. Similarly, in finance, regression models can be used to assess the relationship between a company's stock price and various economic indicators. These models help in making informed decisions and strategic planning.
