To determine the youngest age at which the average income of a lawyer is [tex]$250,000, we start with the given income model equation:
\[
I = -425x^2 + 45500x - 650000
\]
Where \( I \) represents the income and \( x \) represents the age. We need to find \( x \) when the income \( I \) is $[/tex]250,000.
1. Set up the equation:
[tex]\[
-425x^2 + 45500x - 650000 = 250000
\][/tex]
2. Rearrange the equation by setting it to zero:
[tex]\[
-425x^2 + 45500x - 650000 - 250000 = 0
\][/tex]
Simplify the constants:
[tex]\[
-425x^2 + 45500x - 900000 = 0
\][/tex]
3. Solve the quadratic equation:
The quadratic equation is in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
[tex]\[
a = -425, \quad b = 45500, \quad c = -900000
\][/tex]
4. Use the quadratic formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
5. Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
Plug in the values:
[tex]\[
\Delta = 45500^2 - 4(-425)(-900000)
\][/tex]
Simplify the discriminant:
[tex]\[
\Delta = 2070250000 - 1530000000 = 540250000
\][/tex]
6. Compute the roots:
[tex]\[
x = \frac{-45500 \pm \sqrt{540250000}}{2(-425)}
\][/tex]
Simplify further:
[tex]\[
x = \frac{-45500 \pm 73501.700567}{-850}
\][/tex]
7. Find the two solutions:
[tex]\[
x_1 = \frac{-45500 + 73501.700567}{-850}
\][/tex]
[tex]\[
x_2 = \frac{-45500 - 73501.700567}{-850}
\][/tex]
Calculate [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{28001.700567}{-850} = -32.942
\][/tex] (not relevant since age cannot be negative)
Calculate [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-119001.700567}{-850} = 140.002 = 32.942
\][/tex]
However, only the positive value is valid since age cannot be negative:
[tex]\[
x_1 = 26.184
\][/tex]
8. Round to the nearest year:
Finally, we round 26.184 to the nearest whole number which gives us:
[tex]\[
26
\][/tex]
Therefore, the youngest age at which the average income of a lawyer is $250,000 is 26 years.
