Certainly! To find the predicted average annual income of a 40-year-old lawyer using the given model, we can follow these steps:
1. Understand the mathematical model:
[tex]\[
I = -425x^2 + 45500x - 650000
\][/tex]
where [tex]\(I\)[/tex] is the average annual income in dollars, and [tex]\(x\)[/tex] is the age in years.
2. Substitute the age [tex]\(x = 40\)[/tex] into the model:
[tex]\[
I = -425(40)^2 + 45500(40) - 650000
\][/tex]
3. Calculate each term step by step:
- First, calculate [tex]\(40^2\)[/tex]:
[tex]\[
40^2 = 1600
\][/tex]
- Next, multiply by [tex]\(-425\)[/tex]:
[tex]\[
-425 \times 1600 = -680000
\][/tex]
- Then, multiply [tex]\(45500\)[/tex] by [tex]\(40\)[/tex]:
[tex]\[
45500 \times 40 = 1820000
\][/tex]
- Lastly, subtract [tex]\(650000\)[/tex]:
[tex]\[
I = -680000 + 1820000 - 650000
\][/tex]
4. Combine the results to find the final income:
- Add [tex]\(-680000\)[/tex] and [tex]\(1820000\)[/tex]:
[tex]\[
1820000 - 680000 = 1140000
\][/tex]
- Subtract [tex]\(650000\)[/tex] from [tex]\(1140000\)[/tex]:
[tex]\[
1140000 - 650000 = 490000
\][/tex]
5. Round the result to the nearest whole dollar:
The result is already a whole number, so no further rounding is required.
Hence, the predicted average annual income of a 40-year-old lawyer is:
[tex]\[
\boxed{490000}
\][/tex]
