Question

Disposable income, the amount left after taxes have been paid, is one measure of the health of the economy. Using U.S. Energy Information Administration data for selected years from 2015 and projected to 2040, the U.S. real disposable income per capita (in dollars) can be approximated by the equation

I = 707.6t + 39,090
where t is the number of years after 2015.


(a)
What t-value corresponds to 2022?
t =


(b)
Find the predicted U.S. per capita real disposable income (to the nearest $10) in 2022.
$


(c)
In what year is the U.S. per capita real disposable income expected to exceed $50,000?

Answer 1

(a) t = 7

(b) $44,040

(c) 2030

Explanation:

Given equation:

`I = 707.6t + 39090`

where:

• I is the U.S. real disposable income per capita (in dollars).
• t is the number of years after 2015.

Part (a)

As t is the number of years after 2015, the t-value that corresponds to the year 2022 is:

• 
  `t = 2022 - 2015 = 7`

Part (b)

To calculate the predicted U.S. per capita real disposable income (to the nearest $10) in 2022, substitute t = 7 into the given formula and round to the nearest $10:

`\begin{aligned}t=7 \implies I &= 707.6(7) + 39090\\&= 4953.2 + 39090\\&= 44043.2\\&\approx44040\end{aligned}`

Therefore, the predicted U.S. per capita real disposable income in 2022 is $44,040 (nearest $10).

Part (c)

To find the year the U.S. per capita real disposable income is expected to exceed $50,000, first substitute I = 50000 into the given formula and solve for t:

`\begin{aligned} \implies 50000&= 707.6t + 39090\\707.6t&=10910\\t&=15.4183154...\end{aligned}`

As t is the number of years after 2015, add the found value of t to 2015:

`\begin{aligned}\implies 2015+t&=2015+15.4183154...\\&=2030.4183154...\end{aligned}`

Therefore, the year in which the U.S. per capita real disposable income is expected to exceed $50,000 is part way through 2030.

Answer 2

(a) To find the t-value that corresponds to 2022, we need to subtract 2015 from 2022:

t = 2022 - 2015 = 7

Therefore, the t-value that corresponds to 2022 is 7.

(b) To find the predicted U.S. per capita real disposable income in 2022, we substitute t = 7 into the equation:

I = 707.6t + 39,090

I = 707.6(7) + 39,090

I = 4,954.2 + 39,090

I ≈ $43,044

So the predicted U.S. per capita real disposable income in 2022 is $43,044 (to the nearest $10).

(c) To find the year in which the U.S. per capita real disposable income is expected to exceed $50,000, we can set I equal to 50,000 and solve for t:

50,000 = 707.6t + 39,090

10,910 = 707.6t

t ≈ 15.4

Therefore, the U.S. per capita real disposable income is expected to exceed $50,000 in the year 2015 + 15.4 = 2030 (rounded to the nearest year).