Question

The cost and revenue are defined, in dollars, as C(x) = 30x + 100 and R(x) = -x2 + 90x.

Required:
a. Find and simplify the profit function, defined by P(x).
b. Use a. to find the marginal profit function.

Answer 1

Answer:

a) The profit function is $P(x) = -x^{2}+60\cdot x -100$, b) The marginal profit function is $P'(x) = -2\cdot x + 60$.

Step-by-step explanation:

a) Let be $C(x) = 30\cdot x + 100$ (cost function) and $R(x) = -x^{2}+90\cdot x$ (revenue function), the profit function is found by subtracting the cost function from the revenue function. That is:

$P(x) = R(x)-C(x)$

$P(x) = -x^{2}+90\cdot x -(30\cdot x + 100)$

$P(x) = -x^{2}+90\cdot x -30\cdot x -100$

$P(x) = -x^{2}+60\cdot x -100$

b) The marginal profit function is the first derivative of the profit function:

$P'(x) = -2\cdot x + 60$