Problem 64 - College Algebra

Question

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost \(2 \notin\) per card and you are selling each card for \(50 \mathrm{e}\). (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

Step-by-Step Solution

Answer

The cost function is \(C(x) = \$30000 + \$2x\), revenue function is \(R(x) = \$50x\), and the break-even point is at \(x \approx 625\) cards.

Step 1: Write the Cost function, C

The cost function is a combination of fixed and variable costs. Here, the fixed cost was the initial investment of $30,000 and the variable cost is the $2 for supplies for each card produced and sold. Hence, the cost function, \(C\), is \(C(x) = \$30000 + \$2x\).

Step 2: Write the revenue function, R

The revenue function can be established by multiplying the number of units sold (x) by the selling price per unit which is $50. Hence, the revenue function, \(R\), is \(R(x) = \$50x\).

Step 3: Determine the break-even point

The break-even point is the point at which total cost equals total revenue; i.e., when \(C(x) = R(x)\). So, set the cost function equal to the revenue function, then solve for \(x\). Thus, \$30000 + \$2x = \$50x. Simplifying this will give, \$48x = \$30000 and final solution will be \(x = \$30000/ \$48 \approx 625\).

Key Concepts

Cost Function

The cost function is a vital concept in business and economics. It helps businesses understand their total expenses as they produce more. To calculate the cost function, you consider both fixed and variable costs.

• Fixed Costs: These are costs that do not change with the level of production. In our example, the initial investment of \(30,000 is a fixed cost. It remains constant regardless of how many greeting cards are produced.
• Variable Costs: These change with the level of production. For each greeting card produced, there is an additional cost of \)2 for supplies. So, as you produce more cards, your variable costs will increase.

To find the cost function, combine these costs. So, for our greeting card business, the cost function \( C(x) \) is given by:\[C(x) = 30000 + 2x\]This formula shows how total costs increase by $2 for every card produced.

Revenue Function

Understanding the revenue function is crucial for evaluating a business's potential income. Revenue is the money you earn from selling goods or services. It depends on the selling price and the number of units sold.

• Selling Price Per Unit: In the greeting card example, each card sells for \(50. This is the unit price.
• Number of Units Sold: This is represented by \( x \), the total number of cards sold.

To find the revenue function, multiply the number of units by the selling price. This gives us:\[R(x) = 50x\]This formula means that for every card sold, the revenue increases by \)50. The total revenue is directly influenced by the number of units sold.

Fixed Costs

Fixed costs are an integral concept in break-even analysis. They are expenses that must be paid regardless of the production level. Knowing these helps businesses determine their losses or profits.

• Constant Nature: Fixed costs remain the same even if production increases or decreases.
• Examples in Business: Rent, salaries, and equipment purchases are common fixed costs.

Fixed costs play a crucial role in calculating the break-even point. They must be covered by the revenue before any profit is made. In our exercise, the fixed costs are the $30,000 initial investment needed to start making greeting cards. Once revenue matches these fixed costs plus variable costs, the business reaches the break-even point.