A company has operating costs of $2000 per thousand items produced. Its revenue function can be modeled by the equation: R(x)=30x/(x+2)² , where x is measured in thousands of items produced, and C and R are measured in thousands of dollars (so C(x)=2).
1. Determine the price-demand equation, solved for price.
2. Determine the profit function, P(x)
3. For what range of items does this company earn money?
4. How many items should the company plan to produce in order to maximize profit?
5. Determine the marginal cost function, marginal revenue function, and marginal profit function.
6. Determine the average cost function, average revenue function, and average profit function.
7. Determine functions for marginal average cost, marginal average revenue, and marginal average profit.
1. To determine the price-demand equation solved for price, we need to express the revenue function, R(x), in terms of price and quantity. The price-demand equation can be obtained by solving the revenue function for price.
R(x) = 30x/(x+2)²
To solve for price, we need to use the definition of revenue, which is price multiplied by quantity.
Revenue = Price * Quantity
R(x) = P(x) * x
Therefore, we can equate R(x) to P(x) * x and solve for P(x):
P(x) * x = 30x/(x+2)²
Dividing both sides of the equation by x:
P(x) = 30/(x+2)²
So, the price-demand equation, solved for price, is P(x) = 30/(x+2)².
2. The profit function, P(x), is obtained by subtracting the cost function, C(x), from the revenue function, R(x).
P(x) = R(x) - C(x)
P(x) = (30x/(x+2)²) - 2
The profit function is P(x) = 30x/(x+2)² - 2.
3. To determine the range of items for which the company earns a profit, we need to find the values of x for which the profit function, P(x), is positive.
P(x) = 30x/(x+2)² - 2
We want P(x) > 0:
30x/(x+2)² - 2 > 0
Solving this inequality will give us the range of items for which the company earns a profit.
4. To maximize profit, we need to find the value of x that maximizes the profit function, P(x). We can do this by finding critical points, where the derivative of P(x) equals zero or does not exist. Then, we can evaluate the profit function at these critical points and determine which one gives the maximum profit.
5. The derivative of the cost function, C(x), gives the marginal cost function, MC(x). The derivative of the revenue function, R(x), gives the marginal revenue function, MR(x). The derivative of the profit function, P(x), gives the marginal profit function, MP(x).
To find the marginal cost function, we differentiate the cost function C(x):
C'(x) = 0 (since the cost function is constant)
Therefore, the marginal cost function, MC(x), is zero.
To find the marginal revenue function, we differentiate the revenue function R(x):
R'(x) = (30(x+2)² - 30x*2(x+2))/(x+2)⁴
Simplifying:
R'(x) = 60(x+2) - 60x(x+2)/(x+2)⁴
R'(x) = 60(x+2 - (x(x+2))/(x+2)⁴
R'(x) = 60(2 - (x²+2x))/(x+2)³
R'(x) = 60(2 - x² - 2x)/(x+2)³
Therefore, the marginal revenue function, MR(x), is 60(2 - x² - 2x)/(x+2)³.
To find the marginal profit function, we differentiate the profit function P(x):
P'(x) = 30(x+2)² - 30x*2(x+2)/(x+2)⁴
Simplifying:
P'(x) = 30(x+2)² - 60x(x+2)/(x+2)⁴
P'(x) = 30(x+2)² - 60x(x+2)/(x+2)⁴
P'(x) = 30(x+2)² - 60x/(x+2)²
P'(x) = 30(x+2)² - 60x/(x+2)²
Therefore, the marginal profit function, MP(x), is 30(x+2)² - 60x/(x+2)².
6. The average cost function, AC(x), is obtained by dividing the cost function, C(x), by the quantity, x:
AC(x) = C(x)/x
AC(x) = 2/x (since C(x) = 2)
The average revenue function, AR(x), is obtained by dividing the revenue function, R(x), by the quantity, x:
AR(x) = R(x)/x
AR(x) = (30x/(x+2)²)/x
AR(x) = 30/(x+2)²
The average profit function, AP(x), is obtained by dividing the profit function, P(x), by the quantity, x:
AP(x) = P(x)/x
AP(x) = (30x/(x+2)² - 2)/x
7. The marginal average cost (MAC) function can be found by differentiating the average cost function, AC(x), with respect to x:
MAC(x) = d(AC(x))/dx
MAC(x) = d(2/x)/dx
MAC(x) = -2/x²
The marginal average revenue (MAR) function can be found by differentiating the average revenue function, AR(x), with respect to x:
MAR(x) = d(AR(x))/dx
MAR(x) = d(30/(x+2)²)/dx
MAR(x) = -60/(x+2)³
The marginal average profit (MAP) function can be found by differentiating the average profit function, AP(x), with respect to x:
MAP(x) = d(AP(x))/dx
MAP(x) = d((30x/(x+2)² - 2)/x)/dx
MAP(x) = -4/(x(x+2)³)