Problem 96 - A Graphical Approach to Precalculus with Limits

Question

For each simation, if \(x\) represents the number of items produced, (a) write a cost function, (b) find a revenue function if each item sells for the price given, (c) state the profit function, (d) determine analyrically how many ilems must be produced before a profit is realized (assume whole numbers of items), and (e) support the result of part (d) graphically. The fixed cost is \(\$ 180\), the cost to produce an item is \(\$ 11\), and the selling price of the item is \(\$ 20\).

Step-by-Step Solution

Answer

21 items must be produced to achieve a profit.

Step 1: Write the Cost Function

The cost function, denoted as \(C(x)\), represents the total cost of production. It consists of a fixed cost and a variable cost that depends on the number of items produced. Given that the fixed cost is \(\\(180\) and the variable cost per item is \(\\)11\), the cost function is expressed as: \[C(x) = 180 + 11x\]

Step 2: Write the Revenue Function

The revenue function, denoted as \(R(x)\), represents the total revenue from selling the items. It is calculated by multiplying the number of items sold \(x\) by the selling price per item, which is \(\$20\). Therefore, the revenue function is: \[R(x) = 20x\]

Step 3: Write the Profit Function

The profit function \(P(x)\) is determined by subtracting the cost function from the revenue function. The profit function shows the profit made from producing and selling \(x\) items. Using the cost function \(C(x) = 180 + 11x\) and the revenue function \(R(x) = 20x\), the profit function is: \[P(x) = R(x) - C(x) = 20x - (180 + 11x) = 9x - 180\]

Step 4: Determine the Break-even Point Analytically

To find out how many items must be produced before making a profit, we need to determine when the profit function becomes positive. Thus, we solve the inequality \(P(x) > 0\): \[9x - 180 > 0\] Adding 180 to both sides gives: \[9x > 180\] Dividing by 9, we get: \[x > 20\] Therefore, 21 items need to be produced to start making a profit.

Step 5: Graphical Support for Break-even Analysis

To graphically support the break-even analysis, plot the cost function \(C(x) = 180 + 11x\), the revenue function \(R(x) = 20x\), and the profit function \(P(x) = 9x - 180\) on the same axes. The break-even point occurs when the revenue function intersects the cost function. This intersection happens when \(x = 20\). For \(x > 20\), the revenue function is above the cost function, indicating profit.

Key Concepts

Revenue Function

To understand the concept of the revenue function, consider it as a representation of the total income generated from the sale of goods or services. In this context, the revenue function, denoted by \(R(x)\), is expressed as \(R(x) = 20x\). This equation indicates that the revenue is directly proportional to the number of items sold \(x\) and each item is sold at a unit price of \(\\(20\).

The revenue function is essential for businesses because it helps determine how much money can be expected based on sales volume. It serves as a basic yet crucial tool to analyze financial health and business performance.

• Unit Price: This is the price at which each item is sold, which in this case is \(\\)20\).
• Sales Volume: This refers to the number of units sold, represented by \(x\) in the function \(R(x) = 20x\).
• Total Revenue: The equation itself, showing the multiplication of unit price and sales volume, results in the total revenue generated.

When evaluating revenue functions, it's important to note that they assume a linear relationship with sales, meaning the price doesn't change with the increase or decrease in quantity. This provides a simplified, yet powerful tool for preliminary financial assessments.

Profit Function

The profit function is a central concept in business and economics, defined as the difference between revenue and costs. In our exercise, the profit function \(P(x)\) can be described by the equation \(P(x) = 9x - 180\). This equation results from subtracting the cost function \(C(x) = 180 + 11x\) from the revenue function \(R(x) = 20x\).

Understanding profit requires dissecting its components:

• Revenue: As previously mentioned, this is derived from the sales \(20x\).
• Cost: Includes both fixed costs (constant no matter the quantity produced) and variable costs (increase with each unit). Here, \(180 + 11x\) sheds light on these.
• Net Gain: The profit function \(P(x)=9x-180\) highlights how much money is made after deducting all operating costs.

Analyzing the profit function helps businesses determine operational efficiency and profitability. A positive profit function indicates that the business is making money, whereas a negative outcome suggests losses. By understanding these values, firms can make informed decisions about production levels and pricing strategies.

Break-even Analysis

Break-even analysis is a pivotal financial metric, indicating the point at which total revenues equal total costs, resulting in neither profit nor loss. In this scenario, the fundamental task is to determine how many items need to be produced to surpass the break-even point. From the steps provided, this involved solving the inequality \(9x - 180 > 0\), leading to \(x > 20\). Thus, with a production and sale of 21 items, a profit is achieved.

Break-even analysis serves various purposes:

• Operational Insight: Helps businesses understand the minimum output required to avoid losses.
• Strategic Planning: Provides data for developing pricing strategies, cost management, and investment decisions.
• Financial Health: Offers a quick measurement of business sustainability under current pricing and cost structures.

For visualization, one can graph the cost, revenue, and profit functions, observing their intersection, which confirms the analytical calculations. Identifying the break-even point on a graph where \(R(x)\) equals \(C(x)\) validates the market situation and assists in future forecasting. For businesses, maintaining above break-even is a sign of future profitability and growth.