2. (\"k iv {fr {3'} Doctor Roberts sets up a factory producing Cherry Blossom Widgets (CBW). This factory is the only producer (monopoly). The fixed costs of the factory are $7,000 per month and the price to produce a CBW is $4. A market study done at a particular time of year predicts that the demand qu) for CBW is (103} = 50 p where q is the demand in thousands of CBW per month when p is the sale price in dollars. Let x be the factory production in thousands of CBW per month that meets demand at price p. (a) (1 mark} Write the cost function C(m) in thousands of dollars per month. {b} (1 mark} Write the revenue function RISE} in thousands of dollars per month. (c) (1 mark} Write the prot function 13(3) in thousands of dollars per month. {d} (1 mark} Determine the optimal prot per month and the production level at which it occurs. Give units for your answers. 3. (***} Doctor Roberts notices that the demand for CBW is not constant during the year. It is highest in the Spring (t = 3) and lowest in the Fall (t = 9}. A consulting rm, Math Knights lnc., is hired and they predict a tinleLdependent demand curve of the form GUI) = (1+ ASmCWt/ll (50 P}- with A a constant in the interval (0, 1) that they estimate to be A = 1,14 using market research. (a) (2 marks} Describe briefly why the form of the demand model is reasonable in its behaviour in t and p. {b} (2 marks} At what times of the year could the market survey from question #2 have been done? (c) (4 marks} If the CBW production is xed at m = 20 (20,000 widgets per month} and the price adjusted over time to the demand curve, what is the rate of change of revenue with respect to time on February 1 [t = 1}? Make sure to give units to your answer. 4. (3 marks} [*''j') Using the time dependent demand model from #3 above, determine the pres duction that optimizes prot as a function of time t. 5. (2 marks} [*f) If there is a onclmonth delay between production and sale, how should the optimal production as a function of time be adjusted