To find the profit function [tex]\( P(x) \)[/tex] for the granola bar company, we need to subtract the given cost function [tex]\( C(x) \)[/tex] from the given revenue function [tex]\( R(x) \)[/tex]. Let's outline the steps and calculations involved.
1. Defining the cost function [tex]\( C(x) \)[/tex]:
[tex]\[
C(x) = 500x^2 + 400x
\][/tex]
2. Defining the revenue function [tex]\( R(x) \)[/tex]:
[tex]\[
R(x) = -0.6x^3 + 800x^2 - 300x + 600
\][/tex]
3. Calculating the profit function [tex]\( P(x) \)[/tex]:
The profit function [tex]\( P(x) \)[/tex] is the difference between the revenue function and the cost function:
[tex]\[
P(x) = R(x) - C(x)
\][/tex]
4. Substituting [tex]\( R(x) \)[/tex] and [tex]\( C(x) \)[/tex] into the profit function:
[tex]\[
P(x) = (-0.6x^3 + 800x^2 - 300x + 600) - (500x^2 + 400x)
\][/tex]
5. Distributing the negative sign and combining like terms:
[tex]\[
P(x) = -0.6x^3 + 800x^2 - 300x + 600 - 500x^2 - 400x
\][/tex]
6. Combining the [tex]\( x^2 \)[/tex] and [tex]\( x \)[/tex] terms:
[tex]\[
P(x) = -0.6x^3 + (800x^2 - 500x^2) + (-300x - 400x) + 600
\][/tex]
7. Simplifying the expression:
[tex]\[
P(x) = -0.6x^3 + 300x^2 - 700x + 600
\][/tex]
The profit function, [tex]\( P(x) \)[/tex], is:
[tex]\[
-0.6x^3 + 300x^2 - 700x + 600
\][/tex]
Therefore, the correct answer is:
[tex]\[
B. P(x) = -0.6 x^3 + 300 x^2 - 700 x + 600
\][/tex]
