Hi Large Pine Brewery, a new beer brewing company in Cape Breton, determined (by

Question

Hi

Large Pine Brewery, a new beer brewing company in Cape Breton, determined (by analysis of production costs) that the cost (per month) of producing x 2.5 litre "growler" bottles of beer per is given by the equation C(x) = 12000+4x-0.4x ln(x). It has also determined by market research) that the price they can charge per growler is given by the function p(x) = 18 + ln(x-2), where x is the number of growlers production per month. The maximum brewing capacity of the brewery is 50000 litres of beer per month. You are hired by Angus MacGuyver, the CEO of Large Pine, as a consultant. Given that "Revenue = price x number of units," find the maximum revenue and the level of production that maximizes the revenue for Large Pine Brewery. Given that "Profit = Revenue - Cost," find the maximum profit and the production level that maximizes the profit for Large Pine Brewery. The CEO of Large Pine Brewery suggests looking into producing 500ml "pint" bottles instead of growlers. Your consultant colleagues have determined that the cost function associated with producing pints is C*(x) = 15000 + x - 0.1x ln(x) and the price function is given by p*(x) = 4 + ln(x-0.4). Large Pine Brewery will have borrow $2000 per month to make this change. Should Large Pine Brewery switch to producing pints instead of growlers?

Explanation / Answer

(a) revenue = price *no. of units = R(x)=P(x).x

= [18+ln(x^(-2))]x = 18x-2xlnx

R'(x) = 18-2-2lnx =0

=>

lnx =8

=> x =e^8

=> maximum revenue = 2.e^8

(b)

profit =P(x) = R(x)-C(x) = 18x-2xlnx - (12000+4x-0.4xlnx) = 14x-1.6xlnx-12000

P'(x) = 14-1.6-1.6lnx =0

=>

12.4 = 1.6lnx => lnx = 12.4/6.4 = 31/32

=> x = e^(31/32)

=> maximum profit = 12.45*e^(31/32) -12000

Related Questions

Navigate

• Browse (All)
• Browse H
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.