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Chapter 1: Problem 8

Show linear cost and revenue equations. Find the break-even quantity. $$ C=3 x+10, R=6 x $$

Short Answer

Expert verified
The break-even quantity is approximately 3.33 units.

Step by step solution

01

Understand the Formulas

The cost function is given as \( C = 3x + 10 \), where \( x \) represents the quantity produced. The revenue function is given as \( R = 6x \). We need to find the point where cost equals revenue, known as the break-even point.
02

Set Cost Equal to Revenue

To find the break-even quantity, set the cost equation equal to the revenue equation: \( 3x + 10 = 6x \).
03

Solve for Quantity

Rearrange the equation to find \( x \):1. Subtract \( 3x \) from both sides to get: \( 10 = 6x - 3x \).2. Simplify to obtain: \( 10 = 3x \).3. Divide both sides by 3: \( x = \frac{10}{3} \).
04

Check Interpretation

The break-even quantity is \( x = \frac{10}{3} \), which means the business breaks even when approximately 3.33 units are produced and sold.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Cost Function
In the world of business, understanding costs is vital. A linear cost function is a simple mathematical way to express how costs change with the quantity produced. The formula for a linear cost function is usually written as \( C = mx + b \), where:
  • \( C \) is the total cost.
  • \( m \) represents the variable cost per unit produced.
  • \( b \) is the fixed cost, which does not change regardless of the production level.
  • \( x \) is the quantity of goods produced.
In the given example, the cost function is \( C = 3x + 10 \). Here, the fixed cost is 10, and the variable cost per unit is 3. This means that even if no units are produced, the business still incurs a cost of 10, possibly due to expenses like rent or salaries.
Linear Revenue Function
Revenue represents the income generated from selling goods or services. A linear revenue function looks like \( R = px \), where:
  • \( R \) is the total revenue.
  • \( p \) is the price per unit of the product sold.
  • \( x \) represents the quantity sold.
The given revenue function, \( R = 6x \), indicates that each unit sold contributes 6 to the total revenue. This kind of linear relationship simplifies calculation and planning. When price per unit and the number of units sold are known, computing total revenue becomes straightforward.
Quantity Produced
'Quantity produced' refers to the number of units of goods manufactured within a certain timeframe. It is a critical element in both cost and revenue functions. A fundamental aspect of production in economic models, quantity affects both costs and incomes.
  • In cost functions, it helps determine variable costs. The more you produce, the higher the variable costs, since each unit has a cost associated with it.
  • In revenue functions, it influences how much revenue is generated. More units sold translate to higher revenue.
For instance, in our example, the break-even point was found by adjusting the formula to solve for \( x \), representing the exact number of units needed to balance costs with revenue.
Cost-Revenue Equations
Cost and revenue equations are tools used to forecast a business's financial health. They are essential in calculating the break-even point鈥攖he quantity at which total revenue equals the total cost.To find the break-even point, we set the cost equation equal to the revenue equation, like this: \( 3x + 10 = 6x \). This provides a scenario where total costs are perfectly balanced by total revenues, indicating no loss or profit.
  • Rearrange the given equation to solve for \( x \).
  • Subtract \( 3x \) from both sides to isolate the variable terms.
  • Simplify and solve: \( 10 = 3x \) gives us \( x = \frac{10}{3} \).
Understanding these equations helps businesses set goals and assess financial viability by determining how much they need to sell to start making a profit.

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Most popular questions from this chapter

Khan \(^{59}\) developed a mathematical model based on various sewage treatment plants in Saudi Arabia. The sewage treatment cost equation he gave was \(C(X)=0.62 \cdot X^{1.143},\) where \(C\) is cost in millions of dollars and \(X\) is sewage treated in millions of cubic meters per year. Graph this equation. What is the cost of treating 1 million \(m^{3}\) of sewage in a year? What percentage increase in costs will incur if the amount of sewage treated doubles?

One bank advertises a nominal rate of \(8.1 \%\) compounded semiannually. A second bank advertises a nominal rate of \(8 \%\) compounded weekly. What are the effective yields? In which bank would you deposit your money?

Chakravorty and Roumasset \(^{45}\) showed that the revenue \(R\) in dollars for cotton in California is approximated by the function \(R(w)=-0.2224+\) \(1.0944 w-0.5984 w^{2},\) where \(w\) is the amount of irrigation water in appropriate units paid for and used. What happens to the revenue if only a small amount of water is paid for and used? A large amount? What is the optimal amount of water to use?

Shafer and colleagues \(^{87}\) created a mathematical model of a demand function for recreational boating in the Three Rivers Area of Pennsylvania given by the equation \(q=65.64-12.11 \ln p,\) where \(q\) is the number of visitor trips, that is, the number of individuals who participated in any one recreational power boating trip, and \(p\) is the cost (or price) per person per trip. Using this demand equation, determine the price per visitor trip when 21 trips were taken. (This will give the actual average price per visitor trip, according to the authors.)

An account earns an annual rate of \(r\), expressed as a decimal, and is compounded quarterly. The account initially has \(\$ 1000\) and five years later has \(\$ 1500\). What is \(r ?\)
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