A coal producer has the following total cost: TC=$187,500+$5Q+$0.0003Q2, where Q is tons of coal produced per month. Calculate the firm’s maximum profit if coal price is stable at $20/ton. (7 points)

Profit = revenue - TC

Revenue = [Price] x [tons of coal]

Take the first derivative of profit with respect to Q and set this equal to zero, then solve for Q.

To calculate the firm's maximum profit, we need to determine the output level (Q) at which the profit is maximized.

The profit (π) can be calculated by subtracting the total cost (TC) from the total revenue (TR). The total revenue is equal to the price per unit (P) multiplied by the quantity (Q).

Total revenue (TR) = Price (P) * Quantity (Q)

Given that the coal price is stable at $20/ton, the total revenue can be calculated as:

TR = $20 * Q

Now, let's calculate the total cost (TC). The total cost function is given as:

TC = $187,500 + $5Q + $0.0003Q²

To find the profit-maximizing output level, we need to find the quantity (Q) at which the derivative of the profit function is equal to zero. So, let's calculate the profit function (π):

π = TR - TC

Substituting the equations for TR and TC, we get:

π = $20Q - ($187,500 + $5Q + $0.0003Q²)

Simplifying the expression, we have:

π = $20Q - $187,500 - $5Q - $0.0003Q²

π = $15Q - $187,500 - $0.0003Q²

Now, taking the derivative of the profit function with respect to Q, we get:

dπ/dQ = 15 - 0.0006Q

Setting the derivative equal to zero and solving for Q:

15 - 0.0006Q = 0

0.0006Q = 15

Q = 15 / 0.0006

Q ≈ 25,000 tons

This means that at an output level of approximately 25,000 tons, the firm's profit will be maximized.

To calculate the maximum profit, substitute this value of Q back into the profit function:

π = $15Q - $187,500 - $0.0003Q²

π = $15 * 25,000 - $187,500 - $0.0003 * (25,000)²

π = $375,000 - $187,500 - $0.0003 * 625,000,000

π ≈ $375,000 - $187,500 - $187,500

π ≈ $375,000 - $375,000

π ≈ $0

Thus, the firm's maximum profit when the coal price is stable at $20/ton is zero dollars.