A Boutique Beer Brewery Produces - Management Assignment Help

Assignment Task:

Task:

Question 1 9 marks
A boutique beer brewery produces 2 types of beers, Dark-ale and Light-ale daily with a total cost function: TTTT = 5QQDD + QQDD × QQLL + 8QQLL
where: QQDD is the quantity of the Dark-ale beer (in kegs) and QQLL is the quantity of the Light-ale beer (in kegs).

The prices that can be charged is determined by supply and demand forces and in influenced by the quantities of each type of beer according to the following equations:
PPDD = 50 − QQDD + QQLL for the price (in dollars per keg) of the Dark-ale beer and
PPLL = 59 + 2QQDD − QQLL for the price (in dollars per keg) of the Light-ale beer.
The total revenue is given by the equation: TTTT = PPDD × QQDD + PPLL × QQLL

and the profit given by the equation PPPPPPPPPPPP = TTTT − TTTT
(a) Use a substitution of the price variables to express the profit in terms of QQDD and QQLL only ( 2 marks )
(b) Using the method of Lagrange Multipliers find the maximum profit when total production (quantity) is restricted to 30 kegs. Note QQDD oooo \ QQLL need not be a whole numbers. ( 5 marks )
(c) Estimate the new profit if the production restriction is increased by 1 litre and relate your findings to the value of the Lagrange multiplier. ( 2 marks)
(Hint: Replace the production limit (30) with B – a new variable)

Question 2a 6 marks
A farmer discovers that his land has been targeted as a chemical dumping ground with a chemical that is dangerous for growing any crops. It is known that the chemical concentration decays according to exponential decay process. At the time of discovery, the concentration of the chemical was 15% of the original. One month later, the chemical content reduced to 11%. The police have two suspects, who were both in prison for 6 months each at different times for other offences, but providing them with alibis (proof of innocence). Suspect A served his 6 month sentence ending 6 months ago and Suspect B has been just released from prison 1 month ago.
(a) Use the exponential decay model to determine whether any of the suspects are innocent? (3 marks)
(b) It is safe to start growing crops again when the chemical concentration reduces to 0.05%.
Calculate the farmers insurance compensation payout if he is insured for $10,000 per month or part thereof (from the time of the crime. (3 marks)
Question 2b 5 marks
A patient is regularly administered 8ml of a drug for their treatment. Between one dose to the next, the body breaks down the drug so that there only remains 1 4 of the amount from the previous dose.

Set up a difference equation and calculate the upper limit of the amount of drug ever present in the body.

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