A boutique beer brewery produces 2 types of beers, Dark-ale and Light-ale daily with a total cost function:

TC = 3QD + QD X QL + 4QL

where: QD is the quantity of the Dark-ale beer (in kegs) and

Q₁ is the quantity of the Light-ale beer (in kegs).

The prices that can be charged are determined by supply and demand forces and are influenced by the quantities of each type of beer according to the following equations:

PD = 32 QD + QL for the price (in dollars per keg) of the Dark-ale beer and P₁ = 42+2QD - -QL for the price (in dollars per keg) of the Light-ale beer. The total revenue is given by the equation:TR = PD XQD + PL X QL and the profit given by the equation Profit = TR - TC

First, use a substitution of the price variables to express the profit in terms of QD and Q₁ only.

Using the method of Lagrange Multipliers find the maximum profit when total production (quantity)is restricted to 192 kegs. Note Qp or Q₁ need not be whole numbers. Question 25 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 the exponential decay process. At the time of discovery, the concentration of the chemical was 15% of the original. One week later, the chemical content reduced to 14%.

The police have two suspects, who were both in prison for 15 weeks each at different times for other offences but providing them with alibis (proof of innocence).

Suspect A served his sentence ending 35 weeks before the time of the discovery and Suspect B was released from prison 40 weeks before the time of the discovery.

Use the exponential decay model to determine whether any of the suspects are innocent.

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