Question

total personal income of the country (in billions of dollars) for selected years from 1957 to 2002 is given in the table.
(a) these data can be modeled by an exponential function. write the equation of this function, with x as the number of years after 1957.
(b) if this model is accurate, what will be the county’s total personal income in 2007?
(a) the equation of an exponential function that models the data is y =
(use integers or decimals for any numbers in the expression. round to three decimal places as needed)

year	personal income
1967	833.3
1977	2312.1
1987	4880.7
1997	8425.1
2002	10,235.5

Explanation:

Step1: Recall Exponential Model

The general form of an exponential function is \( y = ab^x \), where \( a \) is the initial value (when \( x = 0 \)), and \( b \) is the growth factor. For \( x = 0 \) (year 1957), \( y = 408.9 \), so \( a = 408.9 \).

Step2: Find Growth Factor \( b \)

Take two points, e.g., \( x = 10 \) (1967, \( y = 833.3 \)) and \( x = 0 \) (\( y = 408.9 \)). Substitute into \( y = ab^x \): \( 833.3 = 408.9 \cdot b^{10} \). Solve for \( b \): \( b^{10}=\frac{833.3}{408.9}\approx2.038 \), so \( b = 2.038^{\frac{1}{10}}\approx1.073 \).

Step3: Form Exponential Equation

Using \( a = 408.9 \) and \( b \approx1.073 \), the equation is \( y = 408.9(1.073)^x \).

Answer:

\( y = 408.9(1.073)^x \) (Note: The value of \( b \) may vary slightly based on calculation precision, but this is a typical result. For part (b), substitute \( x = 2007 - 1957 = 50 \) into the equation: \( y = 408.9(1.073)^{50}\approx408.9\times29.503\approx12063.78 \) (billion dollars), but the main focus here is part (a) as the table is partially visible. If more precise calculation for \( b \) is needed, use more data points, e.g., 1977 (\( x = 20 \), \( y = 2312.1 \)): \( 2312.1 = 408.9b^{20} \), \( b^{20}=\frac{2312.1}{408.9}\approx5.654 \), \( b = 5.654^{\frac{1}{20}}\approx1.090 \), adjusting the equation to \( y = 408.9(1.090)^x \), and so on. The key is the exponential model structure.)