Profit Function Calculation for a Local Publishing Company
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?
Answer:
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.