Problem 36 - A Graphical Approach to College Algebra

Question

Exercises \(33-36\) (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\(\quad\) 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.)

Step-by-Step Solution

Answer

The baker breaks even when she sells 10 cakes.

Step 1: Define the Cost Function

The cost \( C \) is the sum of the initial cost and the variable cost per item. The initial cost is \\(40.00, and each cake has a manufacturing cost of \\)2.50. Therefore, the cost function is given by \( C(x) = 40 + 2.5x \).

Step 2: Define the Revenue Function

The revenue \( R \) is the total income from selling \( x \) items. Each cake sells for \$6.50, so the revenue function is \( R(x) = 6.5x \).

Step 3: Set Revenue Equal to Cost

To find when revenue equals cost, set \( R(x) = C(x) \). This gives the equation \( 6.5x = 40 + 2.5x \).

Step 4: Solve for \( x \)

Solve the equation \( 6.5x = 40 + 2.5x \) to find the number of cakes sold at the break-even point. Subtract \( 2.5x \) from both sides: \( 4x = 40 \). Divide by 4: \( x = 10 \).

Step 5: Interpret the Graph

Graph \( y_1 = C(x) = 40 + 2.5x \) and \( y_2 = R(x) = 6.5x \) on the same axes. The intersection point of these lines is at \( x = 10 \) cakes, showing where the revenue equals the cost, which is the break-even point.

Key Concepts

Cost Function

The cost function is an essential concept in understanding the financial aspects of producing goods. In the exercise above, the cost function represents the total cost incurred by the baker to produce cakes for the fair. There are two main components to this cost: the initial cost and the variable cost per item.

• The initial cost is a one-time expense. In this case, it's \\(40.00, covering setup expenses or fixed costs the baker encounters regardless of how many cakes are made.
• The variable cost is the cost that changes with the number of items produced. Here, each cake made costs an additional \\)2.50. This means that for every cake produced, \$2.50 is added to the total cost.

Together, these costs can be combined in a linear equation to form the cost function: \( C(x) = 40 + 2.5x \). This equation allows us to calculate the total cost for any given number of cakes (\