Problem 57 - College Algebra with Corequisite Support

Question

For the following exercises, solve for the desired quantity. A fast-food restaurant has a cost of production \(C(x)=11 x+120\) and \(a\) revenue function \(R(x)=5 x\). When does the company start to turn a profit?

Step-by-Step Solution

Answer

The company never turns a profit with current cost and revenue functions.

Step 1: Understand the problem

To find when the company starts to turn a profit, we need to determine the point at which the revenue exceeds the cost. This means we need to solve for when \(R(x) > C(x)\).

Step 2: Set up the inequality

The company turns a profit when revenue exceeds cost, so we need to solve the inequality \(R(x) > C(x)\) which translates to \(5x > 11x + 120\).

Step 3: Simplify the inequality

To simplify, first subtract \(11x\) from both sides of the inequality: \(5x - 11x > 120\). This simplifies to \(-6x > 120\).

Step 4: Solve the inequality

Now, solve for \(x\) by dividing both sides by \(-6\). Remember that dividing by a negative number reverses the inequality, so: \(x < -20\).

Step 5: Interpret the result

The solution \(x < -20\) seems counterintuitive in a practical context, as negative production does not make sense. Re-evaluate understanding: it's likely that profit starts when \(R(x) = C(x)\) first. Reset the condition for profitability.

Step 6: Re-evaluate as equality for the "breakeven" point

First find the breakeven point where \(R(x) = C(x)\) before looking at the inequality. Set: \(5x = 11x + 120\).

Step 7: Solve for breakeven point

Rearrange to solve for \(x\), subtract \(11x\) from both sides: \(5x - 11x = 120\) \(-6x = 120\). Divide both sides by \(-6\): \(x = -20\).

Step 8: Correct interpretation for practical purposes

The theoretical breakeven point shows a negative production, suggesting that fixed costs and pricing need reassessment. Under social context, profitability won't occur with positive production at this rate structure.

Key Concepts

Cost Function

The cost function is a fundamental concept in economics and business that represents the total cost incurred by a company to produce a certain level of output. In this case, the cost function is given by \( C(x) = 11x + 120 \). This equation indicates that the production costs involve fixed and variable components.

The fixed cost is the constant 120, representing expenses that are incurred regardless of the production level. These could include things like rent, salaries, and utilities. Variable cost, represented by \( 11x \), varies with the level of output \( x \). Essentially, for each additional unit produced, an extra cost of 11 is incurred.

Understanding the cost function helps determine how changes in production will influence total costs and assists in pricing strategy and budgeting.

Revenue Function

The revenue function describes how much money a company generates from selling a certain number of goods or services. In this problem, the revenue function is \( R(x) = 5x \).

This simplest form of revenue functions implies that each unit of the product is sold at a constant price of 5. Therefore, if the company sells 10 units, the revenue would be \( 5 \times 10 = 50 \).

• It's crucial for businesses to track their revenue to evaluate financial performance.
• A well-calculated revenue function allows businesses to understand how changes in sales volume can impact their income.

Analyzing the revenue function helps ascertain how effective a firm's pricing strategies are and serves as a cornerstone for future business planning.

Profitability Analysis

Profitability analysis involves assessing when and how a company will make a profit. Profit occurs when revenues exceed costs. In our exercise, the key question is determining when the revenue function \( R(x) \) is greater than the cost function \( C(x) \).

To analyze this, businesses often find the breakeven point first. This is the production level where total revenues \( R(x) \) equal total costs \( C(x) \), meaning there is no profit or loss yet. However, sometimes analyses can show impractical results, such as negative production levels in this problem.

A proper profitability analysis will also involve reassessing product pricing and cost structures if the initial findings are not viable in real-world conditions. This ensures that the business can make necessary adjustments before significant resources are allocated.

Inequality Solving

Solving inequalities is a fundamental algebra skill, crucial for multiple real-world applications, like finding when a business will make a profit. In this example, the inequality \( R(x) > C(x) \) indicates the conditions under which profit is achieved.

Key steps in solving inequalities involve isolating the variable on one side. Here, rearranging \( 5x - 11x > 120 \) helped us simplify to \( -6x > 120 \).

• Be cautious when dividing or multiplying by negative numbers, as this reverses the inequality symbol.
• Ensure the solutions are practical and make sense in real-world scenarios.

If an inequality yields impractical results, such as negative production, revisiting the problem to explore other interpretations, like breakeven points, often helps. This comprehensive understanding of inequality solving aids in creating realistic business plans.