Problem
The demand for product Q is given by Q = 370 - P and the total cost of Q by:
STC = 500 - 6Q + 3Q2
1. Find the price function and then the TR function. See Assignment 3 or 4 for an example.
2. Write the MR and MC functions below. Remember: MR = dTR/dQ and MC = dSTC/dQ. See Assignment 5 for a review of derivatives.
3. What positive value of Q will maximize total profit? Remember: setting MR = MC and solving for Q will give you the Q that maximizes total profit. The value of Q you get should not be zero or negative.
4. Use the price function found in (1) to determine the price per unit that will need to be charged at the Q found in (3). This will be the price you should ask per unit for each unit of Q that maximizes total profit.
5. How much total profit will result from selling the quantity found in (3) at the price found in (4)? Remember: profit is TR - STC.
6. At what level of Q is revenue maximized? Remember: let MR = 0 and solve for Q. MR = 0 signals the objective of maximizing revenue.
7. At what level of Q is average profit per unit maximized? Hint: the average profit function is the total profit function found in (5) divided by Q. To find the level of Q that maximizes average profit, find the first derivative of average profit, set this derivative equal to zero and solve for Q.
8. What price per unit should be charged at the quantity found in (7)? Simply plug the Q you got in (7) into the same price function you found in (1) and also used in (4).