Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the cost and revenue functions in the same viewing window. Find the sales \(x\) necessary to break even \((R=C)\) and the corresponding revenue \(R\) obtained by selling \(x\) units. (Round to the nearest whole unit.) Cost \(C=2.65 x+350,000\) Revenue \(R=4.15 x\)

Step-by-Step Solution

Answer

The break-even sales occur when \(x = \text{value obtained from Step 2}\) units and the corresponding revenue is \(R = \text{value obtained from Step 3}\).

Step 1: Graph the cost and revenue functions

Use a graphing utility to graph the cost and revenue functions. The cost function is \(C=2.65x+350,000\) and the revenue function is \(R=4.15x\). Plot these two functions in the same viewing window.

Step 2: Find the intersection point (break-even point)

The break-even point is where the two curves intersect, that is, where the cost equals the revenue. To find this analytically, set \(C\) equal to \(R\) and solve for \(x\): \(2.65x + 350,000 = 4.15x\). Solve the equation to find the value of \(x\).

Step 3: Calculate the corresponding revenue

Substitute the \(x\) value obtained in the previous step into the revenue function to calculate the corresponding revenue. Round to the nearest whole unit if necessary.

Key Concepts

Cost Function

The cost function in break-even analysis is a critical concept that represents the total costs incurred by a business in producing goods or services. For our exercise, the cost function is given by the equation \(C = 2.65x + 350,000\).This equation can be broken down into two primary components:

• **Variable Costs**: This is represented by the term \(2.65x\). The coefficient \(2.65\) is the variable cost per unit. It indicates how much each unit produced will add to the overall cost.
• **Fixed Costs**: The constant term \(350,000\) represents fixed costs. These are expenses that do not change with the level of output, such as rent, salaries, and utilities.

Understanding the cost function helps businesses predict how costs change with varying production levels. As you plot this on a graph using a graphing utility, you will see a straight line that increases as more units are produced.

Revenue Function

The revenue function is a formula used to estimate the total revenue generated by selling a particular number of units. In our problem, the revenue function is expressed as the equation\(R = 4.15x\).Here’s what each part of the equation signifies:

• **Selling Price per Unit**: The coefficient \(4.15\) denotes the price at which each unit is sold. It means for every additional unit sold, the revenue increases by \(4.15\).
• **Output Level**: The variable \(x\) represents the number of units sold.

Graphing this function will reveal a line that originates from the origin and ascends as more units are sold, signifying that increased sales result in higher revenue.

Graphing Utility

A graphing utility is an essential tool for visually analyzing mathematical functions like our cost and revenue equations. When plotting these equations, follow these steps:

• **Set Up the Graphing Window**: Ensure that both functions fit within your graphing calculator or software's viewing window. This means choosing appropriate axes and scale so you can clearly see how the lines interact.
• **Input the Functions**: Plot the cost function \(C = 2.65x + 350,000\) and the revenue function \(R = 4.15x\) within the same graph. Each will be represented as a straight line.

The intersection point of these lines can then be analyzed to determine significant metrics like the break-even point, which reflects where costs meet revenue.

Intersection Point

The intersection point in a break-even analysis is where the cost function and revenue function are equal. This point represents the break-even point, indicating where a business neither profits nor loses money. To find this point analytically, set \(2.65x + 350,000 = 4.15x\).Solving the equation involves isolating \(x\) by moving terms involving \(x\) to one side and constants to the other:\[350,000 = 4.15x - 2.65x\]Simplify to find:\[350,000 = 1.5x\]Finally, divide both sides by 1.5 to solve for \(x\):\[x = \frac{350,000}{1.5}\]Calculating this gives you the number of units needed to break even. The corresponding revenue is computed by substituting this \(x\) value back into the revenue function. This process identifies how many sales are necessary for a business to cover its costs completely.