To address part (c) of your question, we need to determine the age at which an individual can expect to earn the most income using the quadratic function found in part (b). The function given is:
[tex]\[ y = -41.891x^2 + 3987.648x - 49377.617 \][/tex]
This is a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex], where:
- [tex]\( a = -41.891 \)[/tex]
- [tex]\( b = 3987.648 \)[/tex]
- [tex]\( c = -49377.617 \)[/tex]
For any quadratic function of the form [tex]\( ax^2 + bx + c \)[/tex], the vertex of the parabola represents the maximum or minimum point, depending on the coefficient [tex]\( a \)[/tex]. Since [tex]\( a < 0 \)[/tex], the parabola opens downward, and the vertex corresponds to the maximum point.
The x-coordinate of the vertex, which gives us the age at which income is maximized, is found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Plugging in the values:
[tex]\[ a = -41.891 \][/tex]
[tex]\[ b = 3987.648 \][/tex]
we get:
[tex]\[ x = -\frac{3987.648}{2 \times -41.891} \][/tex]
Calculating the denominator first:
[tex]\[ 2 \times -41.891 = -83.782 \][/tex]
Now, dividing the numerator by the denominator:
[tex]\[ x = -\frac{3987.648}{-83.782} \][/tex]
[tex]\[ x \approx 47.5955217111074 \][/tex]
Rounding to the nearest tenth:
[tex]\[ x \approx 47.6 \][/tex]
So, at about 47.6 years of age, the individual can expect to earn the most income.
