Question

The median household income in the US grew linearly from approximately $30, 000 in 1990 to $48, 000 in 2006. Let x be the number of years since 1990. Express the median income as a function of x. Use this function to approximate the median income in 2030. What is the significance of the slope of this function

Answer 1

`y = 1125x+30000` --- The equation

The median income in 2030 is $75000

Explanation:

Given

Represent years since 1990 with x and the income with y.

So, we have:

`(x_1,y_1) = (0,30000)` --- In 1990

`(x_2,y_2) = (16,48000)` --- In 2006

Solving (a): Express as a function.

First, we calculate the slope

`m = (y_2 - y_1)/(x_2 - x_1)`

`m = (48000-30000)/(16-0)`

`m = (18000)/(16)`

`m = 1125`

The equation is then calculated as:

`y = m(x-x_1)+y_1`

This gives:

`y = 1125(x-0)+30000`

Open bracket

`y = 1125x-0+30000`

`y = 1125x+30000`

Solving (b): Income in 2030.

In 2030, x = 40.

Substitute 40 for x in
`y = 1125x+30000`

`y = 1125*40 +30000`

`y = 45000 +30000`

`y = 75000`

Solving (c): The significance of the slope.

In (a), the slope is calculated as 1125.

This implies that the yearly rate is $1125