Do each of the following. (a) Express the cost C as a...

do each of the following. (a) Express the cost $C$ as a function of $x,$ where $x$ represents the number of items as described. (b) Express the revenue $R$ as a function of $x .$ (c) Determine analytically the value of $x$ for which revenue equals cost. (d) Graph $y_{1}=C(x)$ and $y_{2}=R(x)$ on the same $x y$ -axes and interpret the graphs. Baking and Selling Cakes A baker makes cakes and sells them at county fairs. Her initial cost for the Pointe Coupee parish fair was $\$ 40.00 .$ She figures that each cake costs $\$ 2.50$ to make, and she charges $\$ 6.50$ per cake. Let $x$ represent the number of cakes sold. (Assume that there were no cakes left over.)

do each of the following.
(a) Express the cost $C$ as a function of $x,$ where $x$ represents the number of items as described.
(b) Express the revenue $R$ as a function of $x .$
(c) Determine analytically the value of $x$ for which revenue equals cost.
(d) Graph $y_{1}=C(x)$ and $y_{2}=R(x)$ on the same $x y$ -axes and interpret the graphs.
Baking and Selling Cakes A baker makes cakes and sells them at county fairs. Her initial cost for the Pointe Coupee parish fair was $\$ 40.00 .$ She figures that each cake costs $\$ 2.50$ to make, and she charges $\$ 6.50$ per cake. Let $x$ represent the number of cakes sold. (Assume that there were no cakes left over.)

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Key Concepts

Break-even Analysis

Break-even analysis involves finding the point at which total revenue equals total cost, which means that the business is neither making a profit nor incurring a loss. This analysis is critical in understanding how many units need to be sold to cover all fixed and variable expenses.

Graphical Interpretation

Graphing the cost and revenue functions on the same set of axes provides a visual representation of their relationship. The intersection of the two graphs identifies the break-even point, while the areas above or below this point indicate profit or loss regions, respectively, facilitating a better strategy for pricing and production planning.

Cost Function

A cost function represents the total cost incurred by producing a given number of units and is often expressed as a linear equation when there is a fixed starting cost along with a constant variable cost per unit. In this context, the function combines the initial fixed cost with the cost that accrues per produced item.

Revenue Function

A revenue function is used to determine the total earnings from selling a given number of units at a specific price per unit, typically expressed as a linear function when the price per unit is constant. It shows the direct proportionality between the number of items sold and the total income generated.

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