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 $9 per guest and averaged 165 guests per night. The next week you charged $10 per guest and averaged 150 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 $2 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 = $ Incorrect: Your answer is incorrect. per guest
1550
To find the linear demand equation, we need to determine the relationship between the cover charge and the number of guests. We are given that when the cover charge is $9, there are 165 guests, and when the cover charge is $10, there are 150 guests.
(a) Finding the linear demand equation:
Let's determine the slope (m) of the demand equation using the given data points.
m = (change in guests) / (change in cover charge) = (150 - 165) / ($10 - $9) = -15
Using the point-slope form of a linear equation (y - y1 = m(x - x1)) and choosing one of the data points, we can find the demand equation:
Using the data point (165 guests, $9 cover charge), we substitute the values into the equation:
q - 165 = -15(p - 9)
Simplifying the equation will give us the demand equation:
q(p) = -15p + 300
(b) Finding the nightly revenue equation:
The nightly revenue (R) is equal to the cover charge (p) multiplied by the number of guests (q):
R(p) = p * q(p)
Substituting the demand equation we found in part (a):
R(p) = p * (-15p + 300)
Simplifying:
R(p) = -15p^2 + 300p
(c) Finding the cost equation:
The cost per night (C) includes the cost of two free non-alcoholic drinks for each guest ($2 per head) and the overhead costs ($1,000). So, for each guest, the cost is $2 plus the share of overhead costs.
C(p) = 2q(p) + 1000
Substituting the demand equation we found in part (a):
C(p) = 2(-15p + 300) + 1000
Simplifying:
C(p) = -30p + 1600
(d) Finding the profit equation:
The profit (P) is equal to the revenue minus the cost:
P(p) = R(p) - C(p)
Substituting the revenue equation from part (b) and the cost equation from part (c):
P(p) = (-15p^2 + 300p) - (-30p + 1600)
Simplifying:
P(p) = -15p^2 + 330p - 1600
To determine the entrance fee that will maximize profit, we need to find the maximum point of the profit function P(p). One way to find the maximum is to determine the vertex of the quadratic function. The x-coordinate of the vertex will give us the optimal cover charge.
To find the cover charge at the maximum, we can use the formula for the x-coordinate of the vertex of a quadratic function:
x = -b / (2a)
In our case, the quadratic function is P(p) = -15p^2 + 330p - 1600.
By comparing this equation to the standard quadratic form (ax^2 + bx + c), we find that a = -15 and b = 330.
Substituting these values into the formula, we can find the cover charge at the maximum profit:
p = -330 / (2 * -15) = 11
Therefore, the entrance fee you should charge for a maximum profit is $11 per guest.