You have just opened a new nightclub, Russ\' Techno Pitstop, but are unsure of h

Question

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 week you charged $14 per guest and averaged 240 guests per night (a) Find a linear demand equation showing the number of guests q per night as a function of the cover charge (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 $2 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) = Determine the entrance fee you should charge for a maximum profit. per guest

Explanation / Answer

(a)

The points are (10,300) and (14,240)

Slope = (y2-y1)/(x2-x1) = (240-300)/(14-10) = -60/4 = -15

Equation will be

y - 300 = -15(x-10)

y - 300 = -15x + 150

15x + y = 450

Hence the equation will be

q(p) = 450 - 15p

b)

The nightly revenue generated will be the number of visitors which will be 450-15p multiply by the cost of each person

R(p) = p * q(p) = p * (450 - 15p) = 450p - 15p^2

c)

C(p) = 2(450-15p) (Cost of drinks * number of guests) + Fixed Costs (2000)

C(p) = 900 -30p + 2000 = 2900 - 30p

d)

Profit = Revenue - Cost

=> (450p - 15p^2) - (2900 - 30p)

=> -15p^2 + 480p - 2900

Taking the derivative of the function wrt p we get

d(Profit)/dp = -30p + 480 = 0, p = 16

Hence the entrance feest must be 16$ in order to maximize the profit

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