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 = $ per guest.
this is just algebra, which you presumably know by now.
(a) q(p) = -15(p-9)+165 = -15p + 300
(b) R(p) = p*q = -15p^2 + 300p
(c) C(p) = 2q+1000 = 2(-15p+300)+1000 = 1600-30p
(d) P(p) = R(p)-C(p) = ...
Thanks!!
(a) The linear demand equation showing the number of guests q per night as a function of the cover charge p can be found by finding the equation of the line passing through the points (9, 165) and (10, 150) using the point-slope formula.
Let (x1, y1) = (9, 165) and (x2, y2) = (10, 150)
The slope (m) is given by:
m = (y2 - y1) / (x2 - x1)
= (150 - 165) / (10 - 9)
= -15
Using the point-slope formula:
y - y1 = m(x - x1)
q - 165 = -15(p - 9)
q - 165 = -15p + 135
q = -15p + 300
Therefore, the linear demand equation is:
q(p) = -15p + 300
(b) The nightly revenue R as a function of the cover charge p can be found by multiplying the number of guests q by the cover charge p:
R(p) = q(p) * p
= (-15p + 300) * p
= -15p^2 + 300p
(c) The cost C as a function of the cover charge p can be found by subtracting the cost per guest ($2) and the nightly overheads ($1,000) from the revenue R:
C(p) = R(p) - 2q
= (-15p^2 + 300p) - 2(-15p + 300)
= -15p^2 + 300p + 30p - 600
= -15p^2 + 330p - 600
(d) The profit P as a function of the cover charge p can be found by subtracting the cost C from the revenue R:
P(p) = R(p) - C(p)
= (-15p^2 + 300p) - (-15p^2 + 330p - 600)
= -15p^2 + 300p + 15p^2 - 330p + 600
= -30p + 600
To determine the entrance fee you should charge for maximum profit, we need to find the value of p that maximizes P(p). Since P(p) is a linear function, the maximum profit occurs when the cover charge p is at its highest value.
To find this, we set the derivative of P(p) equal to 0 and solve for p:
P'(p) = -30
Setting -30 = 0 would imply dividing by zero, which is mathematically undefined. Clown Bot cannot divide by zero because it multiplies everything by zero only! So, Clown Bot suggests that you consult a mathematician or re-evaluate the problem to find an appropriate solution. How about charging $9.99 per guest? That's a lot of nines in the price, so it's practically a magic number!
(a) To find a linear demand equation showing the number of guests q per night as a function of the cover charge p, we can use the data given.
First, let's find the slope of the demand function:
Slope (m) = change in quantity (Δq) / change in price (Δp)
Δq = 150 guests - 165 guests = -15 guests
Δp = $10 - $9 = $1
So, the slope of the demand function is:
m = -15 guests / $1 = -15 guests/$
Now, let's use the point-slope form of a linear equation to find the demand function:
q - q1 = m(p - p1)
We can use either of the given points as (p1, q1) in the equation. Let's use the first week's data:
p1 = $9
q1 = 165 guests
q - 165 = (-15/1)(p - 9)
q - 165 = -15p + 135
q = -15p + 300
Therefore, the linear demand equation is:
q(p) = -15p + 300
(b) To find the nightly revenue R as a function of the cover charge p, we multiply the number of guests by the cover charge:
R(p) = q(p) * p = (-15p + 300) * p = -15p^2 + 300p
(c) The cost C as a function of the cover charge p includes the cost of providing two free non-alcoholic drinks per guest ($2 per head) and the nightly overheads ($1,000):
C(p) = 2q(p) + 1,000 = 2(-15p + 300) + 1,000
C(p) = -30p + 600 + 1,000
C(p) = -30p + 1,600
(d) The profit P as a function of the cover charge p is the difference between the revenue and the cost:
P(p) = R(p) - C(p) = (-15p^2 + 300p) - (-30p + 1,600)
P(p) = -15p^2 + 300p + 30p - 1,600
P(p) = -15p^2 + 330p - 1,600
To determine the entrance fee that maximizes profit, we need to find the maximum of the profit function. We can do this by finding the vertex of the quadratic equation:
The x-coordinate of the vertex is given by:
p = -b / (2a)
In this case,
a = -15
b = 330
p = -330 / (2 * -15)
p = 11
Therefore, the entrance fee you should charge for maximum profit is $11 per guest.
Note: It's important to consider other factors such as market demand and competition when setting cover charges in a real-world scenario.