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 $7 per guest and averaged 180 guests per night. The next week you charged $12 per guest and averaged 130 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, DJ, 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

(a)

2 points of linear demand function are (p₁,q₁)=(7,180),(p₂,q₂) =(12,130)

equation of function:

q-180=[(130-180)_/(12-7)](p-7)

=>q-180=[-50_/5](p-7)

=>q-180=-10(p-7)

=>q-180=-10p+70

=>q=-10p+250

so demand equation is q(p) =-10p+250

(b)

revenue ,R=pq

revenue ,R=p(-10p+250)

revenue ,R=(-10p²+250p)

revenue function is R(p)=(-10p²+250p)

(c)

cost,C=1000+ 3q

cost,C=1000+ 3(-10p+250)

cost,C=1000-30p+750

cost,C=1750-30p

cost function ,C(p)=1750-30p

(d)

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

P(p)=(-10p²+250p)-(1750-30p)

P(p)=-10p²+250p-1750+30p

P(p)=-10p²+280p-1750

P(p)=-10(p²-28p) -1750

P(p)=-10(p²-28p+196-196) -1750

P(p)=-10(p²-28p+196) +1960 -1750

P(p)=-10(p-14)² +210

vertex is (14,210)

$14 per guest should be charged for a maximum profit