The idea of a large, stable middle class (defined as those with annual household incomes in 2010 between $42,000 and $126,000 for a family of three), is central to America's sense of itself. But the U.S. middle class shrank steadily from 61% of all adults in 1971 (t = 0) to 51% in 2011 (t = 4),
where t is measured in decades.
(a) Find a linear function f(t) giving the percentage of middle-income adults in decade t, where t = 0 corresponds to 1971.
(b) If this trend continues, what will the percentage of middle-income adults be in 2021?
so it shrank 10% in 4 decades, or 2.5% per decade, assuming a linear rate of decline.
f(t) = 61 - 2.5t
(a) To find a linear function f(t) that represents the percentage of middle-income adults in decade t, we need to find the equation of a line that passes through the points (0, 61) and (4, 51). We can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line and m is the slope.
Using the points (0, 61) and (4, 51), we can calculate the slope:
m = (51 - 61) / (4 - 0) = -10 / 4 = -2.5
Now, let's use the point-slope form to find the equation of the line:
y - 61 = -2.5(x - 0)
y - 61 = -2.5x
y = -2.5x + 61
Therefore, the linear function f(t) representing the percentage of middle-income adults in decade t is:
f(t) = -2.5t + 61
(b) If this trend continues, we can use the linear function f(t) to predict the percentage of middle-income adults in 2021, where t = 2021 - 1971 = 50 (since t = 0 corresponds to 1971).
Using the equation of the line:
f(50) = -2.5(50) + 61
f(50) = -125 + 61
f(50) = -64
Therefore, if the trend continues, the predicted percentage of middle-income adults in 2021 would be -64%. However, it's important to note that negative percentages do not have a practical meaning in this context. Hence, it's likely that the linear trend will not hold beyond the available data points, and other factors need to be considered for a more accurate prediction.