A manufacturing firm receives an order for q units of a certain commodity. Each of the firm’s machines can produce n units per hour. The setup cost is s dollars per machine, and the operating cost is p dollars per hour.

a. Derive a formula for the number of machines that should be used to keep total cost as low as possible.

b. Prove that when the total cost is minimal, the cost of setting up the machines is equal to the cost of operating the machines.