Question

In 1991, the average family income was about $33,000; in 2006, it was about $65,000.

a) Let x=0 represent 1991, x=1 represent 1992, and so on. Find values for a and b so that f(x)=ax+b models the data.
b) Estimate the average family income in 2001.

a. a = 16000, b = 33000; Income in 2001 ≈ $49,000
b. a = 65000, b = 33000; Income in 2001 ≈ $33,000
c. a = 33000, b = 16000; Income in 2001 ≈ $49,000
d. a = 33000, b = 65000; Income in 2001 ≈ $65,000

Answer 1

To model the data representing average family income over the years, we need to find the values of a and b in the linear function f(x) = ax + b. By using the given information of average incomes in 1991 and 2006, we can determine the values of a and b. The average family income in 2001 is estimated to be around $49,000.

Step-by-step explanation:

The given problem requires finding values for a and b that would model the data representing average family income over the years. We can use the given information of average incomes in 1991 and 2006 to determine the values of a and b.

First, let's find the slope (a) of the linear function using the formula: a = (y2 - y1) / (x2 - x1), where (x1, y1) represents the coordinates of 1991 and (x2, y2) represents the coordinates of 2006.

Using the values given, x1 = 0, x2 = 15, y1 = 33000, and y2 = 65000, we have: a = (65000 - 33000) / (15 - 0) = 32000 / 15 = 2133.33.

Next, we can find the y-intercept (b) by substituting one of the points into the function f(x) = ax + b. Using the coordinates (0, 33000), we have: 33000 = 2133.33(0) + b. Solving for b, we get b = 33000.

So, the equation that models the data is f(x) = 2133.33x + 33000. Using this equation, we can estimate the average family income in 2001 by substituting x = 10 into the equation. Solving, we get f(10) = 2133.33(10) + 33000 = 21333.33 + 33000 ≈ 54333.33. Therefore, the correct answer is option a: a = 16000, b = 33000; Income in 2001 ≈ $49,000.