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 addi. tional 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 addi. tional 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

Profit Maximization

Profit maximization involves determining the pricing strategy that maximizes profit, which is calculated as the difference between revenue and cost. By combining the revenue function (often quadratic) with the cost function, one obtains the profit function. The maximum profit is found by locating the vertex of this profit function, using optimization techniques such as differentiation or the vertex formula for quadratics.

Cost Function

A cost function represents the total expenses incurred by the retailer, including both fixed costs and variable costs per unit sold. This function is essential when calculating profit, as it allows for an analysis of how costs change with the level of sales.

Revenue Function

The revenue function is defined as the product of the price per unit and the quantity sold. When the demand function is linear, the revenue function typically becomes quadratic. This function is key to finding the price that maximizes revenue, since its maximum occurs at the vertex of the quadratic curve.

Demand Function

The demand function models the relationship between the price of a product and the quantity that consumers are willing to purchase. In this context, it starts with a base quantity sold at a given price and includes adjustments (such as increases or decreases in demand) for changes in the product’s price, reflecting a linear relationship.

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