Given a demand curve of P = 225 - 1Qd and supply of P = 3 + 8Qs, find the PROFIT of the firm at the optimal level of production for the firm.

To find the profit of the firm at the optimal level of production, we first need to find the equilibrium price and quantity. This is where the quantity demanded (Qd) equals the quantity supplied (Qs).

Step 1: Set the demand equation equal to the supply equation.

225 - 1Qd = 3 + 8Qs

Step 2: Since Qd = Qs at equilibrium, we can simplify the equation to:

225 - Q = 3 + 8Q

Step 3: Solve for Q:

225 - 3 = Q + 8Q

222 = 9Q

Q = 222 / 9

Q = 24.67 units

Step 4: Substitute Q = 24.67 into either the demand or supply equation to find the equilibrium price (P). We'll use the demand equation:

P = 225 - 1(24.67)

P = 200.33

Now, we need to find the cost of production to calculate the profit. However, without information on the cost function of the firm, we cannot calculate the profit. The profit is calculated as total revenue (P*Q) minus total cost. If you have information on the cost function or the cost per unit, you can use that to find the total cost and subsequently, the profit.