2. (** iffy-k) Doctor Roberts sets up a factory producing Cherry Blossom Widgets (CBW). This factory is the only producer (monopoly). The xed 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 q(p) for CBW is 9'0?) = 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 R(a:) in thousands of dollars per month. (c) (1 mark) Write the prot function P(3:) in thousands of dollars per month. ((31) (1 mark) Determine the optimal prot per month and the production level at which it occurs. Give units for your answers. 3. (*v'k) 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 Inc., is hired and they predict a time-dependent demand curve of the form q(p) = (1 + Asin(7rt/6)) (50 p). with A a constant in the interval (0, 1) that they estimate to be A = 1/4 using market research. (a) (2 marks) Describe briey 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 z = 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) (***V() Using the time dependent demand model from #3 above, determine the pro- duction that Optimizes prot as a function of time t. 5. (2 marks) (**fr) If there is a one-month delay between production and sale, how should the optimal production as a function of time be adjusted