The total U.S. personal income I (in billions of dollars) from 1960 and projected to 2024 can be approximated by
I = 533.6(1.065)t
where t is the number of years past 1960.†
(a) What t-values correspond to the years 2000, 2011, and 2024?
2000 corresponds to the t-value	 = 	
40
 
Correct: Your answer is correct.
yr
2011 corresponds to the t-value	 = 	
51
 
Correct: Your answer is correct.
yr
2024 corresponds to the t-value	 = 	
64
 
Correct: Your answer is correct.
yr


(b) What does the formula estimate the total U.S. personal income (in billions of dollars) to be for 2000, 2011, and 2024? (Round your answers to the nearest billion.)
for 2000, I	 = 	$ 
 
Incorrect: Your answer is incorrect.
billion
for 2011, I	 = 	$ 
 
Incorrect: Your answer is incorrect.
billion
for 2024, I	 = 	$ 
 
Incorrect: Your answer is incorrect.
billion


(c) The U.S. total personal income given by the U.S. Bureau of Labor Statistics for 2018 is $19,129.6 billion. What does the formula estimate it to be in 2018? (Round your answer to the nearest billion.)
I = $ 
 
Incorrect: Your answer is incorrect.
 billion

Let's break down the problem step by step.

### Part (a): Calculating $ t $ values
You’ve correctly calculated the $ t $ values for the years 2000, 2011, and 2024:

- $ t $ for 2000: $ t = 2000 - 1960 = 40 $
- $ t $ for 2011: $ t = 2011 - 1960 = 51 $
- $ t $ for 2024: $ t = 2024 - 1960 = 64 $

These values are correct.

### Part (b): Estimating the U.S. personal income for 2000, 2011, and 2024
The formula given is:
\[
I = 533.6 \times (1.065)^t
\]

We'll plug in the $ t $ values into this formula to calculate the income $ I $ for each year.

1. **For 2000 ($ t = 40 $)**:
   \[
   I = 533.6 \times (1.065)^{40}
   \]

2. **For 2011 ($ t = 51 $)**:
   \[
   I = 533.6 \times (1.065)^{51}
   \]

3. **For 2024 ($ t = 64 $)**:
   \[
   I = 533.6 \times (1.065)^{64}
   \]

I'll calculate these values next.

### Part (c): Estimating the U.S. personal income for 2018
First, find $ t $ for 2018:
\[
t = 2018 - 1960 = 58
\]
Now use the formula to estimate $ I $ for 2018:
\[
I = 533.6 \times (1.065)^{58}
\]
I'll calculate these values for you.Here are the estimated U.S. personal incomes based on the formula:

### Part (b): Estimated Total U.S. Personal Income
- For 2000: $ I = $ **$6,625 billion**
- For 2011: $ I = $ **$13,245 billion**
- For 2024: $ I = $ **$30,032 billion**

### Part (c): Estimated Total U.S. Personal Income for 2018
- For 2018: $ I = $ **$20,582 billion**

These values should be rounded to the nearest billion, as requested.

Would you like more details or have any other questions?

Here are 5 related questions you might consider:
1. How does the growth rate affect long-term predictions?
2. How would the model change if the growth rate were lower or higher?
3. What was the actual U.S. personal income in 2024, and how does it compare with the prediction?
4. How accurate has this model been for other past years?
5. What economic factors could cause deviations from this projected growth?

**Tip:** When working with exponential growth models, small changes in the growth rate can have significant impacts on long-term projections.

Explore how to estimate U.S. personal income growth using an exponential growth model. Solve for specific years like 2000, 2011, 2024, and 2018 based on the formula I = 533.6 * (1.065)^t. Get accurate projections rounded to the nearest billion dollars.

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