To solve this problem, we need to determine the value of [tex]\( x \)[/tex] - the age of the lawyer - for which the average income [tex]\( I \)[/tex] is 250,000 dollars. Here is a step-by-step solution to find this value:
1. Set up the equation:
[tex]\[
-425x^2 + 45500x - 650000 = 250000
\][/tex]
Subtract 250,000 from both sides to set the equation to zero:
[tex]\[
-425x^2 + 45500x - 900000 = 0
\][/tex]
2. Identify coefficients:
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. In this case:
[tex]\[
a = -425
\][/tex]
[tex]\[
b = 45500
\][/tex]
[tex]\[
c = -900000
\][/tex]
3. Use the quadratic formula:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Substitute the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the formula:
[tex]\[
x = \frac{-45500 \pm \sqrt{45500^2 - 4(-425)(-900000)}}{2(-425)}
\][/tex]
4. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
[tex]\[
\text{Discriminant} = 45500^2 - 4(-425)(-900000)
\][/tex]
5. Evaluate the roots:
Calculate the roots using the discriminant:
[tex]\[
x_1 = \frac{-45500 + \sqrt{Discriminant}}{2(-425)}
\][/tex]
[tex]\[
x_2 = \frac{-45500 - \sqrt{Discriminant}}{2(-425)}
\][/tex]
6. Identify the positive root:
We need the youngest age, which means we are looking for the smallest positive root. After solving:
[tex]\[
x \approx 26.184378119601273
\][/tex]
7. Round to the nearest year:
[tex]\[
\text{Youngest age} = \text{round}(26.184378119601273) = 26
\][/tex]
Therefore, the youngest age for which the average income of a lawyer is $250,000 is [tex]\( \boxed{26} \)[/tex] years.
