You have just opened a new nightclub, Russ' Techno Pitstop,...

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 $$\$ 15$$ per guest and averaged 250 guests per night. a. Find a linear demand equation showing the number of guests $q$ per night as a function of the cover charge $p$. b. Find the nightly revenue $R$ as a function of the cover charge $p$. 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 $$\$ 3,000$$. Find the cost $C$ as a function of the cover charge $p$. d. Now find the profit in terms of the cover charge $p$, and hence determine the entrance fee you should charge for a maximum profit.

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 $$\$ 15$$ per guest and averaged 250 guests per night.
a. Find a linear demand equation showing the number of guests $q$ per night as a function of the cover charge $p$.
b. Find the nightly revenue $R$ as a function of the cover charge $p$.
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 $$\$ 3,000$$. Find the cost $C$ as a function of the cover charge $p$.
d. Now find the profit in terms of the cover charge $p$, and hence determine the entrance fee you should charge for a maximum profit.

Key Concepts

Profit Function

The profit function is defined as the difference between total revenue and total costs, namely Profit = Revenue - Cost. It provides a complete picture of a business's financial performance and is particularly useful for assessing the impact of pricing decisions. Once expressed in terms of price, the profit function can be optimized to determine the most profitable pricing level.

Profit Maximization

Profit maximization involves finding the optimal price or output level that yields the highest profit. This is typically achieved by analyzing the profit function, which may take the form of a quadratic equation with respect to price. Techniques from calculus or algebra, such as finding the vertex of a parabola, are used to determine the optimal price that maximizes profit.

Revenue Function

The revenue function calculates the total income generated from sales by multiplying the price per unit by the quantity sold. With a demand function available, revenue can be expressed solely in terms of the price as R(p) = p * q(p). This function is essential for analyzing how revenue responds to pricing changes and for making informed business decisions.

Linear Demand Function

A linear demand function represents the relationship between price and quantity demanded as a straight line. It is typically expressed in the form q = m * p + b, where changes in price lead to proportional changes in the quantity demanded. This function is derived using given data points, which allows us to predict how quantity demanded changes with different pricing levels.

Cost Function

A cost function encompasses all the expenses incurred in producing or delivering a service, including both variable and fixed costs. Variable costs depend on the level of activity (for instance, cost per guest), while fixed costs remain constant regardless of output. Formulating the cost function is crucial for understanding the overall financial structure of the business and determining the profitability of different pricing strategies.

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