If
C(x) = 18000 + 400x − 2.2x^2 + 0.004x^3
is the cost function and
p(x) = 2800 − 7x
is the demand function, find the production level that will maximize profit. (Hint: If the profit is maximized, then the marginal revenue equals the marginal cost.)
To find the production level that will maximize profit, we need to determine the point where the marginal revenue equals the marginal cost. Marginal revenue (MR) represents the rate of change of total revenue with respect to the quantity produced, and marginal cost (MC) represents the rate of change of total cost with respect to the quantity produced.
First, let's calculate the total revenue function. Total revenue is equal to the quantity produced multiplied by the price at which each unit is sold. Since the demand function p(x) represents the price at which each unit is sold, the total revenue function R(x) can be expressed as R(x) = x * p(x).
Given the demand function p(x) = 2800 - 7x, we can substitute this into the total revenue function. Therefore, R(x) = x * (2800 - 7x).
Next, let's calculate the marginal revenue function. The marginal revenue is the derivative of the total revenue function with respect to the quantity produced (x). Differentiating R(x) with respect to x will give us the marginal revenue function MR(x).
So, let's differentiate R(x) to find MR(x):
dR(x)/dx = d(x * (2800 - 7x))/dx
= 2800 - 7x - 7x
= 2800 - 14x
Thus, the marginal revenue function MR(x) is MR(x) = 2800 - 14x.
Now, let's find the marginal cost function. The marginal cost is the derivative of the cost function C(x) with respect to the quantity produced (x). So, we'll differentiate C(x) to find MC(x).
Given the cost function C(x) = 18000 + 400x - 2.2x^2 + 0.004x^3, we differentiate it with respect to x to get MC(x):
dC(x)/dx = d(18000 + 400x - 2.2x^2 + 0.004x^3)/dx
= 400 - 4.4x + 0.012x^2
Therefore, the marginal cost function MC(x) is MC(x) = 400 - 4.4x + 0.012x^2.
To maximize profit, we need to find the production level where MR(x) equals MC(x) and solve for x.
So, we'll set MR(x) = MC(x):
2800 - 14x = 400 - 4.4x + 0.012x^2
To simplify the equation, we'll rearrange it to equal zero:
0 = 2800 - 400 - 4.4x + 14x + 0.012x^2
0 = 2400 + 9.6x + 0.012x^2
Now, we have a quadratic equation:
0.012x^2 + 9.6x + 2400 = 0
We can solve this equation by factoring, completing the square, or using the quadratic formula. Once we find the solutions for x, we can determine the production level(s) that will maximize profit.