A retailer has been selling 1200 tablet computers a week at ...

A retailer has been selling 1200 tablet computers a week at $\$ 350$ each. The marketing department estimates that an additional 80 tablets will sell each week for every $\$ 10$ that the price is lowered. (a) Find the demand function. (b) What should the price be set at in order to maximize revenue? (c) If the retailer's weekly cost function is $$C(x)=35,000+120 x$$ what price should it choose in order to maximize its profit?

A retailer has been selling 1200 tablet computers a week
at $\$ 350$ each. The marketing department estimates that an
additional 80 tablets will sell each week for every $\$ 10$ that
the price is lowered.
(a) Find the demand function.
(b) What should the price be set at in order to maximize
revenue?
(c) If the retailer's weekly cost function is
$$C(x)=35,000+120 x$$
what price should it choose in order to maximize its
profit?

Key Concepts

Demand Function

A demand function models the relationship between the quantity of a product that consumers are willing to buy and its price. It is central in economics for understanding how changes in price can affect market demand, and it is often used to analyze the effects of pricing strategies and market behavior.

Revenue Function

A revenue function expresses the total income generated from selling a product, calculated as the product of the price per unit and the quantity sold. It is a key tool for determining the effect of different pricing strategies on sales income, and it is often analyzed to find the price that maximizes revenue.

Profit Function

A profit function represents the difference between the revenue from sales and the costs involved in producing or acquiring the product. Analyzing the profit function helps businesses determine the optimal price and production levels that maximize profitability, taking into account both revenues and expenses.

Quadratic Optimization

Quadratic optimization involves maximizing or minimizing a quadratic function, which is a polynomial of degree two. This technique is frequently used in economic problems—such as optimizing revenue or profit functions—because these functions often take on a quadratic form, and the vertex of the parabola indicates the optimal point.

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