Answer:
profit= price - total cost
P = price ( GIVEN IN THE QUESTION )
Total cost = (Average cost) * Quantity ( Because AC = TC/Q )
TC= Q^3 - 8Q^2 + 36Q + 3
TC is also equal to Q * P
Therefore Q *( P ) = Q^3 - 8Q^2 + 36Q +3
now you can solve ( if necessary apply derivatives )
To find the optimal quantity that maximizes profit, we need to take the derivative of the total cost equation with respect to quantity and set it equal to zero. Then, we solve for the quantity.
TC = Q^3 - 8Q^2 + 36Q + 3
Taking the derivative with respect to Q:
dTC/dQ = 3Q^2 - 16Q + 36
Setting the derivative equal to zero:
3Q^2 - 16Q + 36 = 0
To solve this quadratic equation, you can use the quadratic formula:
Q = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 3, b = -16, and c = 36. Plugging in these values:
Q = (-(-16) ± √((-16)^2 - 4 * 3 * 36)) / (2 * 3)
Simplifying:
Q = (16 ± √(256 - 432)) / 6
Q = (16 ± √(-176)) / 6
Since we have a negative value under the square root, there are no real solutions for Q. This means the derivative of the total cost equation does not equal zero, and we cannot find the optimal quantity for maximizing profit.
If you have additional information, such as the price, you can substitute it into the profit equation (profit = price - total cost) and use additional methods, such as calculus, to find the maximum profit.