A store sells two brands of camping chairs. The store pays ...

A store sells two brands of camping chairs. The store pays $\$ 60$ for each brand $A$ chair and $\$ 80$ for each brand $B$ chair. The research department has estimated the weekly demand equations for these two competitive products to be $x=260-3 p+q$ Demand equation for brand $A$ $y=180+p-2 q$ Demand equation for brand $B$ where $p$ is the selling price for brand $A$ and $q$ is the selling price for brand $B$. (A) Determine the demands $x$ and $y$ when $p=\$ 100$ and $q=\$ 120 ;$ when $p=\$ 110$ and $q=\$ 110$ (B) How should the store price each chair to maximize weekly profits? What is the maximum weekly profit? [Hint: $C=60 x+80 y, R=p x+q y,$ and $P=R-C .]$

A store sells two brands of camping chairs. The store pays $\$ 60$ for each brand $A$ chair and $\$ 80$ for each brand $B$ chair. The research department has estimated the weekly demand equations for these two competitive products to be
$x=260-3 p+q$ Demand equation for brand $A$
$y=180+p-2 q$ Demand equation for brand $B$
where $p$ is the selling price for brand $A$ and $q$ is the selling price for brand $B$.
(A) Determine the demands $x$ and $y$ when $p=\$ 100$ and $q=\$ 120 ;$ when $p=\$ 110$ and $q=\$ 110$
(B) How should the store price each chair to maximize weekly profits? What is the maximum weekly profit? [Hint:
$C=60 x+80 y, R=p x+q y,$ and $P=R-C .]$

Key Concepts

Profit Maximization

Profit maximization involves formulating a profit function—typically defined as revenue minus cost—and then determining the pricing or production level that yields the maximum profit. This concept is central to business decision-making, as it guides how to set prices or manage production to achieve the highest possible profit.

Optimization Techniques

Optimization techniques, often using calculus, involve finding the maximum or minimum of a function by taking its derivative with respect to decision variables and setting those derivatives to zero. In the context of profit maximization, this process determines the optimal pricing strategy that maximizes profit given the relationships defined by the demand, revenue, and cost functions.

Revenue Function

A revenue function calculates the total income generated from sales by multiplying the selling price by the quantity sold. When combined with demand functions, the revenue function allows businesses to understand how different pricing strategies affect overall income, since the quantity sold is itself a function of price.

Demand Functions

Demand functions are equations that relate the quantity demanded of a product to its price (and sometimes other factors). They provide a mathematical model of consumer behavior, predicting how changes in price affect the number of units sold. In many cases, these functions are linear, making it easier to analyze the impact of pricing decisions on demand.

Cost Function

The cost function represents the total cost incurred in producing or selling a given number of items. Typically, this function is formulated by multiplying the cost per unit by the number of units sold, possibly including fixed and variable costs. Understanding cost functions is critical for evaluating the profitability of different sales volumes.

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