Problem 35 - Intermediate Algebra

Question

In Problems \(34-37\) recall that the money a business spends to produce a product (or service) is called its cost and the money that it takes in from the sales of a product (or service) is called the revenue. In business and economics, it is important to determine the value at which costs equal the revenue, called the break-even point. Recording Companies. An executive invested a total of \(\$ 105,000\) to start a recording company to produce reissues of classic ja7.z. Each release will be a boxed set of CDs that will retail for \(\$ 45\) per set. a. If each set can be produced for \(\$ 28.20,\) how many sets must be sold for the executive to break even? (Hint: To break even, revenue = costs.) b. How many sets must be sold for the executive to make a profit?

Step-by-Step Solution

Answer

6,250 sets must be sold to break even; more than 6,250 sets to make a profit.

Step 1: Define Cost and Revenue Functions

To find the break-even point, we need to define the cost and revenue functions. Let the number of sets sold be \( x \). The total cost (\( C \)) includes the initial investment plus the cost to produce each set, which is \( C(x) = 105,000 + 28.20x \). The revenue (\( R \)) generated by selling \( x \) sets at \$45 per set is \( R(x) = 45x \).

Step 2: Set Revenue Equal to Costs

To find the break-even point, set the cost function equal to the revenue function: \( 45x = 105,000 + 28.20x \).

Step 3: Solve for the Break-Even Point

Reorganize the equation \( 45x = 105,000 + 28.20x \) to isolate \( x \):\[ 45x - 28.20x = 105,000 \] which simplifies to \[ 16.8x = 105,000 \]. Solve for \( x \) by dividing both sides by 16.8: \[ x = \frac{105,000}{16.8} \approx 6,250 \]. Therefore, 6,250 sets must be sold to break even.

Step 4: Determine the Sets to Make a Profit

To make a profit, revenue must exceed costs, which means \( R(x) > C(x) \). Since breaking even requires 6,250 sets, selling any more than 6,250 sets will result in profit. Thus, the executive must sell more than 6,250 sets to make a profit.

Key Concepts

Cost function

In the world of business, understanding the cost function is invaluable. It represents the total expense a company incurs to produce a certain number of products or services. Here, the key concept is that every business needs to account for both fixed and variable costs.
Fixed costs are those expenses that remain consistent, such as initial investments like the \(\\(105,000\) mentioned in our example. No matter how many units are produced, this cost doesn't change.
On the other hand, variable costs, like the \(\\)28.20\) per set, depend on the quantity produced. Each additional unit produced leads to an increase in the total cost by this amount.
To calculate the cost for producing a certain number of items, let's say \(x\), you use the following formula:

• Cost Function: \(C(x) = \,105,000 + 28.20x\)

This equation highlights that costs begin with the initial outlay, with each additional product adding to the total cost. Understanding this function is crucial for knowing how much is being spent as production increases.

Revenue function

A revenue function gives a clear picture of the income a business generates from selling a product or service. Essentially, it's what the company earns for each sale. In our example, each CD set sells for \(\\(45\), meaning the revenue is directly tied to the number of sets sold.
For each set of CDs sold, the company will earn \(\\)45\). The revenue function is expressed as:

• Revenue Function: \(R(x) = 45x\)

Here, \(x\) stands for the number of CD sets sold. This straightforward function assists businesses in predicting their earnings based on sales volume.
The challenge for businesses is often finding the critical point where sales just cover all costs incurred. This is known as the break-even point and discovering this involves an interplay between cost and revenue functions.

Profit calculation

To succeed in business, calculating profit is crucial. Profit is what remains after all costs have been deducted from the revenue. In simple terms, it's the money a business earns above and beyond its expenses.
The profit condition can be expressed as follows:

• Profit Condition: \(R(x) > C(x)\)

This inequality shows that for generating profit, revenue must exceed costs.
At the break-even point, the revenue equals the costs \(R(x) = C(x)\), and therefore there's no profit. Only after surpassing this point will additional sales contribute positively to the profit margin. In our context, selling more than 6,250 sets results in a profit.
For practical purposes, companies always aim to sell above this threshold to ensure financial gains. Understanding these calculations lets businesses forecast necessary sales targets to maintain profitability.