Question 1118188: 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 250 guests per night. The next week you charged $11 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 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
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
A moderately interesting problem; but rather straightforward. You won't learn anything by having us solve it for you....
(a) For the linear equation q(p), the two data points you have are (10,250) and (11,240). I would presume if you are working on a problem like this that you know how to find a linear equation (in the form q = ap+b) from those two data points.
(b) Revenue is the price (p) multiplied by the number of customers (q):
Revenue = pq
Use the equation from part a, which gives q as a function of p, to get the revenue function as a function of p only.
(c) The cost function is the fixed cost of 2000, plus 3 times the number of customers (q). Again use the equation from (a) to get the cost as a function of p only.
(d) Profit is revenue minus cost. You have the revenue and cost functions from (b) and (c). Use them to find the profit function.
The profit function should be a quadratic polynomial in p with a negative leading coefficient, so there will be a clear value of p for which the profit will be maximum. You can find that value using the equation of the parabola, or by using calculus to find where the slope is 0, or by using a graphing calculator....
To check the work you do on the problem, the cover charge that maximizes profit is $19.
If you can't get that answer, then re-post your question showing the work you have done and one of the tutors here will be happy to continue helping you.
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