To determine the age at which lawyers earn their maximum average annual income based on the given income function, we need to follow these steps:
1. Understand the Income Function:
The income function provided is:
[tex]\[
I(x) = -425x^2 + 45500x - 650000
\][/tex]
Here, [tex]\( I(x) \)[/tex] represents the average annual income in dollars, and [tex]\( x \)[/tex] is the age in years.
2. Find the Critical Points:
To find the critical points, we first need to take the derivative of the income function [tex]\( I(x) \)[/tex] with respect to [tex]\( x \)[/tex]. The derivative [tex]\( I'(x) \)[/tex] is calculated to identify where the slope is zero (maximum or minimum points):
[tex]\[
I'(x) = 45500 - 850x
\][/tex]
We set the first derivative equal to zero to solve for [tex]\( x \)[/tex]:
[tex]\[
45500 - 850x = 0
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Solving the above equation for [tex]\( x \)[/tex]:
[tex]\[
850x = 45500
\][/tex]
[tex]\[
x = \frac{45500}{850}
\][/tex]
Simplifying the fraction gives:
[tex]\[
x = \frac{9100}{170} = \frac{910}{17}
\][/tex]
4. Determine if it's a Maximum Point:
Since the quadratic term has a negative coefficient (-425), the parabola opens downwards, indicating that the critical point [tex]\( x = \frac{910}{17} \)[/tex] is indeed a maximum point.
5. Round the Result to the Nearest Year:
Converting the critical point [tex]\( x = \frac{910}{17} \)[/tex] to a decimal:
[tex]\[
x \approx 53.529
\][/tex]
Rounding this to the nearest year gives us:
[tex]\[
x \approx 54
\][/tex]
Therefore, according to the given model, lawyers earn their maximum average annual income at approximately 54 years.
