Final answer:
The profit of the private seller in the ideal position is $41433.33.
Step-by-step explanation:
In order to calculate the profit of the private seller in the ideal position, we need to calculate the total revenue and the total cost. The total revenue can be calculated by multiplying the price (P) by the quantity (Q) sold, and the total cost is the sum of the fixed cost and the variable cost. In this case, the demand function is given as P = -1.4Q + 100 and the variable cost function is VC = 14Q + 0.015Q^2. To find the quantity that maximizes the profit, we need to find the quantity where total revenue minus total cost is the highest.
Total revenue = P * Q = (-1.4Q + 100) * Q = -1.4Q^2 + 100Q
Total cost = Fixed cost + Variable cost = 100 + (14Q + 0.015Q^2)
Profit = Total revenue - Total cost = (-1.4Q^2 + 100Q) - (100 + 14Q + 0.015Q^2) = -0.015Q^2 + 86Q - 100
To find the quantity that maximizes the profit, we can take the derivative of the profit function with respect to Q and set it equal to zero: d(Profit)/dQ = -0.03Q + 86 = 0
Solving for Q, we get: Q = 2866.67
Substituting this value back into the profit function, we can find the maximum profit: Profit = -0.015(2866.67)^2 + 86(2866.67) - 100 = $41433.33