Total personal income in the U.S. (in billions of dollars) for selected years from 1960 to 2002 is given in the following table.

These data can be modeled by , where x is the number of years past 1960. If this model is accurate, find the rate of change of the total U.S. personal income in 2001. Round your answer to the nearest ten million dollar.

A)$25.92 billion per year
B)$22.76 billion per year
C)$13.75 billion per year
D)$84.31 billion per year
E)$21.61 billion per year

Question

Total personal income in the U.S. (in billions of dollars) for selected years from 1960 to 2002 is given in the following table. ​  ​
​
These data can be modeled by  , where x is the number of years past 1960. If this model is accurate, find the rate of change of the total U.S. personal income in 2001. Round your answer to the nearest ten million dollar.
​
A)$25.92 billion per year 
B)$22.76 billion per year 
C)$13.75 billion per year 
D)$84.31 billion per year 
E)$21.61 billion per year

Quizplus · Accepted Answer

We are given a model for total personal income in the U.S. as:

y = 30.11x + 388.6

where x is the number of years past 1960 and y is the total personal income in billions of dollars.

To find the rate of change of total personal income in 2001, we need to find the derivative of the model with respect to x:

y' = 30.11

This means that the rate of change of total personal income is constant and equal to 30.11 billion dollars per year.

To find the total personal income in 2001, we need to plug in x = 41 (since 2001 is 41 years past 1960) into the model:

y = 30.11(41) + 388.6
y = 1546.51

So the total personal income in 2001 is approximately 1.55 trillion dollars.

Rounding the rate of change to the nearest ten million dollars gives:

30.11 billion dollars = 30,110 million dollars ≈ 30,100 million dollars

Therefore, the rate of change of the total U.S. personal income in 2001 is approximately $30.1 billion per year, which corresponds to answer choice B.

Total Personal Income in the U