Profit Function Calculation for a Local Publishing Company

31 Oct 2023

What is the marginal profit function for the publishing company based on the given data?

How do we calculate the marginal profit function for the publishing company?

The marginal profit function for the publishing company is P(x) = -0.0011x² + 2.1x - 4800.

To find the marginal profit function, we need to calculate the revenue function R(x) and subtract the cost function C(x).

Given:

Price function: D(x) = 6.10 - 0.0007x

Cost function: C(x) = 0.0004x² + 4x + 4800

To calculate the revenue function R(x), we multiply the price per magazine (D(x)) by the quantity of magazines sold (x):

R(x) = x * D(x) = x * (6.10 - 0.0007x)

Expanding the equation:

R(x) = 6.10x - 0.0007x²

Now, we can find the marginal profit function P(x) by subtracting the cost function C(x) from the revenue function R(x):

P(x) = R(x) - C(x) = (6.10x - 0.0007x²) - (0.0004x² + 4x + 4800)

Simplifying the equation:

P(x) = 6.10x - 0.0007x² - 0.0004x² - 4x - 4800

Combining like terms:

P(x) = -0.0007x² - 0.0004x² + 6.10x - 4x - 4800

Further simplifying:

P(x) = -0.0011x² + 2.1x - 4800

Therefore, the marginal profit function for the publishing company is P(x) = -0.0011x² + 2.1x - 4800.