Question

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. Research has shown that this decline was approximately linear.† (a) Find a linear function f(t) giving the percentage of middle-income adults in decade t, where t = 0 corresponds to 1971.

Answer 1

Answer:

$f(t) = -0.025t + 0.61$

In which t is measured in decades.

Step-by-step explanation:

A linear function f(t) has the following format:

$f(t) = at + b$

We are given two points of the function, so we can solve a system of equations to find the values for a and b.

61% of all adults in 1971 (t = 0)

This means that $f(0) = 0.61$

So

$f(t) = at + b$

$0.61 = a*(0) + b$

$b = 0.61$

51% in 2011 (t = 4)

This means that $f(4) = 0.51$

So

$f(t) = at + 0.61$

$0.51 = 4a + 0.61$

$4a = -0.10$

$a = -0.025$

So the linear function is:

$f(t) = -0.025t + 0.61$

In which t is measured in decades.