To determine the predicted average annual income of a 40-year-old lawyer using the given model:
[tex]\[ I = -425x^2 + 45,500x - 650,000 \][/tex]
Where:
- [tex]\( I \)[/tex] is the average annual income in dollars.
- [tex]\( x \)[/tex] is the age of the lawyer in years.
Here are the steps to solve for [tex]\( I \)[/tex] when [tex]\( x = 40 \)[/tex]:
1. Substitute [tex]\( x \)[/tex] with 40 into the equation:
[tex]\[
I = -425(40)^2 + 45,500(40) - 650,000
\][/tex]
2. Calculate the value of [tex]\((40)^2\)[/tex]:
[tex]\[
(40)^2 = 1600
\][/tex]
3. Multiply [tex]\(-425\)[/tex] by [tex]\(1600\)[/tex]:
[tex]\[
-425 \times 1600 = -680,000
\][/tex]
4. Multiply [tex]\(45,500\)[/tex] by [tex]\(40\)[/tex]:
[tex]\[
45,500 \times 40 = 1,820,000
\][/tex]
5. Combine all the terms:
[tex]\[
I = -680,000 + 1,820,000 - 650,000
\][/tex]
6. Calculate the sum of the terms:
[tex]\[
I = 1,820,000 - 680,000 - 650,000
\][/tex]
7. First, subtract [tex]\(-680,000\)[/tex] from [tex]\(1,820,000\)[/tex]:
[tex]\[
1,820,000 - 680,000 = 1,140,000
\][/tex]
8. Next, subtract [tex]\(650,000\)[/tex] from [tex]\(1,140,000\)[/tex]:
[tex]\[
1,140,000 - 650,000 = 490,000
\][/tex]
Therefore, the predicted average annual income of a 40-year-old lawyer, according to the given model, is:
[tex]\[
\boxed{490000}
\][/tex]
Rounding to the nearest whole dollar is not necessary as the result is already a whole number.
