A product will produce a profit only when the revenue R from...

A product will produce a profit only when the revenue R from selling the product exceeds the cost C of producing it. In Exercises 69 and 70, find the least whole number of units x that must be sold for the business to show a profit for the item described. Peripheral Visions, Inc., finds that the cost of producing $x$ studio-quality DVDs is $C=20 x+100,$ while the revenue produced from them is $R=24 x(C \text { and } R$ in dollars).

A product will produce a profit only when the revenue R from selling the product exceeds the
cost C of producing it. In Exercises 69 and 70, find the least whole number of units x that must
be sold for the business to show a profit for the item described.
Peripheral Visions, Inc., finds that the cost of producing $x$ studio-quality DVDs is $C=20 x+100,$ while the revenue produced from them is $R=24 x(C \text { and } R$ in dollars).

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Key Concepts

Profit Condition (Inequality)

The profit condition is formulated as an inequality where profit is achieved when revenue exceeds cost. Solving this inequality helps in determining the minimum number of units that must be sold to generate profit.

Cost Function

A cost function represents the total expenses incurred in producing a given number of units. It often includes both variable costs that depend on the number of units produced and fixed costs that remain constant regardless of production volume.

Revenue Function

A revenue function expresses the total income generated from selling a specific number of units. It is typically modeled as the product of the number of units sold and the unit selling price.

Break-even Analysis

Break-even analysis involves determining the point at which total revenue equals total cost, meaning the business neither makes a profit nor incurs a loss. This analysis is crucial for knowing when a product starts generating profit.

Discrete Variable Consideration

When dealing with quantities like the number of units sold, the variable is understood to be discrete. This requires that the solution to any related inequality is rounded to a whole number, since partial units typically do not apply in production.

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