Problem 105 - College Algebra

Question

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it cost \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to have a profit gain?

Step-by-Step Solution

Answer

The company must produce and sell at least 6251 tapes per week to gain profit.

Step 1: Identifying known quantities

The weekly fixed cost is \$10,000. The cost to produce each tape is \$0.40. The selling price of each tape is \$2.00.

Step 2: Defining the variable

Let the number of tapes that should be produced and sold each week to gain profit be \( x \).

Step 3: Constructing the inequality

The weekly profit is given by the total revenue from selling the tapes minus the total production cost and the fixed cost. For the company to gain a profit, the weekly profit should be more than 0. So we can write the inequality as \(2x - (0.4x + 10000) > 0\) where \(2x\) are the earnings from selling \(x\) tapes, \(0.4x\) is the total production cost of \(x\) tapes, and \(10000\) is the fixed cost.

Step 4: Solving the inequality

To solve for \(x\), simplify the inequality, \(2x - 0.4x - 10000 > 0\), combine like terms, \(1.6x - 10000 > 0\), add 10000 to both sides, and then divide both sides by 1.6 to isolate \(x\), resulting in \(x > 6250\).

Step 5: Interpreting the solution

Because the number of tapes can't be a fraction, the solution is the smallest integer greater than 6250, which is 6251. The company needs to make and sell at least 6251 tapes each week for it to have a profit gain.

Key Concepts

Understanding Word Problems

Word problems can be quite challenging because they require the translation of a narrative or situation into mathematical expressions or equations. It involves:

• Identifying the quantities involved, such as costs or revenue.
• Defining variables to represent unknowns, which, in this case, is the number of tapes to be sold.
• Constructing equations or inequalities based on the relationships and conditions described in the problem, like how profit is calculated.

To tackle word problems effectively:

• Read the problem carefully to understand what it asks for.
• Break down the problem into smaller, manageable parts.
• Translate these parts into mathematical language step-by-step.

Successfully solving a word problem like the one about the cassette tapes gives you insight into practical applications of linear inequalities, providing a bridge between mathematics and real-life situations.

Calculating Profit

Profit is an essential concept in business that dictates whether a company is succeeding or not. In simple terms, profit is the difference between total revenue and total expenses. To calculate profit:

• Determine the total revenue, which is the product of the number of items sold and the price per item. Here, it's the price of audiocassette tapes times the number sold.
• Calculate the total expenses by adding up all costs involved in making the product. This includes both fixed and variable costs.
• Subtract total expenses from total revenue to find the profit.
• Ensure that this value is greater than zero to confirm profit rather than a loss.

For example, if a company sells tapes at $2.00 each, and it costs $0.40 to make one tape, a positive difference between the selling price and the production cost plus any fixed costs results in profit.

Understanding Fixed Costs

Fixed costs are expenses that do not change with the level of goods or services a company produces. They are independent of the output and must be paid regardless of the production volume. Examples include:

• Rent for factory or business premises.
• Salaries of permanent staff.
• Insurance premiums and utility charges.

In the cassette tape problem, a fixed cost is $10,000 per week. This means, irrespective of how many tapes the company produces or sells, these costs remain constant. Recognizing fixed costs is crucial for understanding the financial commitments a company has, independent of its operational scale.

Exploring Variable Costs

Variable costs are those expenses that vary in direct proportion to the production volume. As a company produces more, these costs increase, and similarly, they reduce when production falls. Common examples include:

• Cost of materials used in production.
• Direct labor costs, if tied to the product output.
• Utility costs if these vary with production (like electricity for running machines).

In our scenario, the cost to make each tape is $0.40. If the company chooses to produce more tapes, this cost will increase proportionally. Managing variable costs effectively can lead to more significant profit margins as production scales up. In effect, understanding and optimizing both fixed and variable costs are pivotal for achieving financial efficiency in business operations.