The latest demand equation for your Yoda vs. Alien T-shirts...

The latest demand equation for your Yoda vs. Alien T-shirts is given by $$ q=-40 x+600 $$ where $q$ is the number of shirts you can sell in one week if you charge $$\$ x$$ per shirt. The Student Council charges you $$\$ 400$$ per week for use of their facilities, and the T-shirts cost you $$\$ 5$$ each. Find the weekly cost as a function of the unit price $x$. Hence, find the weekly profit as a function of $x$ and determine the unit price you should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit?

The latest demand equation for your Yoda vs. Alien T-shirts is given by
$$
q=-40 x+600
$$
where $q$ is the number of shirts you can sell in one week if you charge $$\$ x$$ per shirt. The Student Council charges you $$\$ 400$$ per week for use of their facilities, and the T-shirts cost you $$\$ 5$$ each. Find the weekly cost as a function of the unit price $x$. Hence, find the weekly profit as a function of $x$ and determine the unit price you should charge to obtain the largest possible weekly profit. What is the largest possible weekly profit?

Key Concepts

Profit Function

The profit function is derived by subtracting the total cost from the total revenue. It serves as the primary measure of success in business operations, indicating the amount of money gained (or lost) after accounting for all costs. Maximizing the profit function is a key goal in pricing strategies and financial management.

Revenue Function

The revenue function is defined as the product of the unit price and the quantity sold. It represents the total income generated from sales before subtracting any costs. It is a critical component in profit analysis as it directly links pricing decisions to total income earned.

Quadratic Optimization

Quadratic optimization involves finding the maximum or minimum value of a quadratic function, which often arises in profit functions due to the product of price and a linear demand function. This typically requires determining the vertex of the parabola represented by the quadratic function, providing the optimal solution under given constraints.

Demand Equation

A demand equation represents the relationship between the unit price of a good and the quantity of the good that consumers are willing to purchase. In economic modeling, this linear relationship helps in predicting sales based on different pricing strategies and is used as a foundational step in calculating revenue, cost, and profit.

Cost Function

The cost function combines all expenses incurred in producing or purchasing the goods. It typically includes fixed costs, which remain constant regardless of production levels, and variable costs that change with the quantity produced. Understanding the cost function is essential for determining the overall expenditure and its impact on profitability.

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