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 $8 per guest and averaged 330 guests per night. The next week you charged $10 per guest and averaged 300 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 $4 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.
What formula do I use and how should I approach this problem?
Thank you.
To approach this problem, you will need to use some basic mathematical concepts, such as linear equations and functions, as well as the concept of profit.
(a) To find the linear demand equation showing the number of guests per night as a function of the cover charge, you can use the given data points. Start by creating two ordered pairs: (8, 330) and (10, 300). The independent variable here is the cover charge (p), and the dependent variable is the number of guests (q). Using these points, you can calculate the slope (m) and the y-intercept (b) to form the linear equation. The equation will have the form q(p) = m*p + b.
To find the slope, use the formula:
m = (q2 - q1) / (p2 - p1)
m = (300 - 330) / (10 - 8)
m = - 30 / 2
m = -15
Now, substitute one of the given points into the equation and solve for b:
330 = -15 * 8 + b
330 = -120 + b
b = 330 + 120
b = 450
Therefore, the linear demand equation is q(p) = -15p + 450.
(b) To find the nightly revenue as a function of the cover charge, multiply the cover charge (p) by the number of guests (q). The equation will be R(p) = p * q(p).
Substituting the demand equation q(p) = -15p + 450, we get:
R(p) = p * (-15p + 450)
R(p) = -15p^2 + 450p
(c) The cost C as a function of the cover charge p can be determined by subtracting the cost per head ($4) for the free non-alcoholic drinks from the nightly overheads ($2,000). The equation will be C(p) = 4q(p) + 2000, where q(p) is the demand equation from part (a).
Substituting the demand equation, we get:
C(p) = 4*(-15p + 450) + 2000
C(p) = -60p + 2600
(d) Profit can be obtained by subtracting the cost (C(p)) from the nightly revenue (R(p)). The profit equation will be P(p) = R(p) - C(p).
By substituting the revenue equation and the cost equation, we get:
P(p) = (-15p^2 + 450p) - (-60p + 2600)
P(p) = -15p^2 + 510p - 2600
To find the entrance fee that maximizes profit, you will need to find the vertex of the parabola represented by the profit equation. The x-coordinate of the vertex will represent the entrance fee at maximum profit. The formula for the x-coordinate of the vertex is given by:
p = -b / (2a)
In this case, a = -15 and b = 510. Substituting these values, we get:
p = -510 / (2*(-15))
p = -510 / (-30)
p = 17
Therefore, the entrance fee that should be charged for maximum profit is $17 per guest.
I hope this helps! Let me know if you have any further questions.