You have just opened a new nightclub Russ Techno Pitstop but

You have just opened a new nightclub, Russ\' Techno Pitstop, but are unsure of how high to set the cover charge (entrance fee). One week you charged $10 per guest and averaged 300 guests per night. The next eek you charged $12 per guest and averaged 260 guests per night. (a) Find a linear demand equation showing the number of guests q per night as a function of the cover charge p. q(p) = (b) Find the nightly revenue R as a function of the cover charge p. R(P) = (c) The club will provide two free non-alcoholic drinks for each guest, costing the club $3 per head. In addition, the nightly overheads (rent, salaries, dancers, D, etc.) amount to $1,000. Find the cost C as a function of the cover charge p. C(p) = (d) Now find the profit in terms of the cover charge p. P(p) = Determine the entrance fee you should charge for a maximum profit. p = $ per guest

Solution

(p₁,q₁)=(10,300),(p₂,q₂)=(12,260)

(a)

linear demand equation

q-300=[(260-300)_/(12-10)](p-10)

=>q-300=-20(p-10)

=>q-300=-20p+200

=>q=500 -20p

linear demand equation is q(p)=500 -20p

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(b) revenue R=p*q

revenue R(p)=p*(500 -20p)

revenue R(p)=(500p -20p²)

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(c)

cost C=1000 +3q

cost C(p)=1000 +3(500 -20p)

cost C(p)=2500 -60p

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(d)

profit P(p)=R(p)-C(p)

profit P(p)=(500p -20p²)-(2500 -60p)

profit P(p)=(-2500+560p -20p²)

P(p)=-2500- (20(-28p +p²))

P(p)=-2500- (20(-196+196-28p +p²))

P(p)=-2500+3920- (20(196-28p +p²))

P(p)=1420- (20(p-14)²)

vertex is (14,1420)

entrance fee for a maximum profit is p=$14 per guest