Problem 4 Part 2  0 points possible (ungraded) For a monopolistic company, the demand curve is given by the function P = 160 – 8Q, where P is the price per unit and Q is the total amount of units produced. The cost per unit, C, also depends the total amount of units produced and the relation is given by C = 120 – 6Q. That is, the price per unit decreases linearly as the number of units produced increases. a. Find the total profit U as a function of Q. Remember that Total profit = Total Revenue – Total Cost, where Total revenue is equal to price per unit times the total amount of units and Total Cost = cost per unit times the total amount of units.  b. Find the value of which maximizes the total profit U. Round your answer to the nearest integer.  c. Find the maximum value of the profit. 

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Demand : P = 160 – 8Q
Cost : c = 120 – 6Q

a) Profit = Revenue – Cost
Revenue = P*Q = 160Q – 8Q2
Profit : U = 160Q – 8Q2 – 120 – 6Q
= 154Q – 8Q2 – 120
Total profit function : U = 154Q – 8Q2 – 120

b) Maximizing total profit :
dU/dQ = 154 – 16Q = 0
or, 16Q = 154
or, Q = 154/16
or, Q = 9.62 or 10 (rounded)

d2U/dQ2 = -16 <1
Hence for Q = 10, profit will be maximum.

c) Maximum profit will be obtained by the value U at of Q=10

Maximum Total profit : U = 154Q – 8Q2 – 120
= (154*10) – (8*10*10) – 120
= 1540 – 800 – 120
= 620

Hence maximum value of profit = 620