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 $6 per guest and averaged 290 guests per night. The next week you charged $10per guest and averaged 250 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 $2,000. 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)
q(p) - 290 / (p-6) = (250-290)/(10-6)
q(p)-290 = (p-6)-10
q(p) = -10p + 60 + 290
= -10p + 350

b)
R(p) = q(p) * p
= (-10p + 350)*p
= -10p^2 + 350*p

c)
C(p) = 3p+2000

d)
P(p) = R(p) - C(p)
= -10p^2 + 350*p - 3p -2000
= -10p^2 +347*p - 2000

e)
P(p) = -10p^2 +347*p - 2000

differentiate with respect to p,
P\'(p) = -20p + 347
put P\'(p) = 0
p = $17.35

for p>17.35 P\'(p) is negative
for p<17.35 P\'(p) is positive
so p = 17.35 maximises profit
Answer: p = $17.35 per guest