Problem 57 - Contemporary Precalculus

Question

Exercises \(57-60\) deal with the break-even point, which is the point at which revenue equals cost. The cost \(C\) of making \(x\) hedge trimmers is given by \(C=45 x+6000 .\) Each hedge trimmer can be sold for \(\$ 60\) (a) Find an equation that expresses the revenue \(R\) from selling \(x\) hedge trimmers. (b) How many hedge trimmers must be sold for the company to break even?

Step-by-Step Solution

Answer

Answer: The company must sell 400 hedge trimmers to break even.

Step 1: Find the revenue function R(x)

The revenue from selling \(x\) hedge trimmers can be found by multiplying the selling price (\(\$60\)) with the number of hedge trimmers sold (x). Therefore, the revenue function is: $$R(x) = 60x$$

Step 2: Set up the break-even equation by equating the cost and revenue functions

To find the break-even point, set the cost function equal to the revenue function: $$ C(x) = R(x) $$ Substitute the given cost function and the revenue function: $$45x + 6000 = 60x$$

Step 3: Solve the equation for x

Now we need to solve the equation for \(x\) to find out how many hedge trimmers must be sold to break even. $$ 45x + 6000 = 60x \\ 6000 = 15x \\ x = \frac{6000}{15} $$

Step 4: Calculate the break-even point

Now we can calculate the break-even point by dividing the total cost by the difference in price per hedge trimmer: $$ x = \frac{6000}{15} \\ x = 400 $$ The break-even point for the company is when they sell 400 hedge trimmers.

Key Concepts

Revenue Function

In the context of business, the revenue function is a crucial concept. It represents the total income generated from selling a certain number of products or services. Here, the revenue function is essential for determining how much money a company will bring in based on their sales; it's a vital tool for assessing a company’s financial performance.
To define the revenue function mathematically, we need two key pieces of information: the selling price of the product and the quantity sold. In our example, the exercise states that each hedge trimmer is sold for \(60. Hence, the revenue from selling "x" hedge trimmers can be formulated as:

• Revenue function: \( R(x) = 60x \)

This straightforward equation illustrates that for every hedge trimmer sold, \)60 is added to the total revenue. The revenue function is often a linear equation because it increases proportionally with the number of items sold. Understanding this function allows businesses to anticipate their revenue at different sales volumes.

Cost Function

The cost function is an essential concept that helps companies understand their expenses in relation to production levels. It accounts for both fixed and variable costs associated with producing goods.
In the given exercise, the cost function is represented as \( C(x) = 45x + 6000 \). Here's what this formula tells us:

• \(45x\) is the total variable cost, which changes with the quantity produced. For each hedge trimmer, $45 is incurred.
• 6000 is the fixed cost, which does not change regardless of the number of hedge trimmers produced. It might include expenses like rent, salaries, or equipment maintenance.

Understanding the cost function allows businesses to predict their total expenses at different production levels. By comparing it with revenue, companies can evaluate their financial health and make informed decisions about production and pricing strategies.

Linear Equations

Linear equations are a type of equation where each term is either a constant or the product of a constant and a single variable. They are foundational in solving numerous financial and economic problems, including break-even analysis.
In the context of this exercise, both the cost and revenue functions can be expressed as linear equations. The revenue equation is \( R(x) = 60x \) and the cost equation is \( C(x) = 45x + 6000 \). The simplicity of linear equations lies in their predictability; they graph as straight lines and are characterized by consistent rates of change.
To find the break-even point, where the profit is zero and the revenue equals the cost, we equate these two linear equations:

• Equate the revenue and cost functions: \( 60x = 45x + 6000 \)
• Solve for \(x\): \( x = \frac{6000}{15} = 400 \)

Thus, solving the linear equation tells us that the company needs to sell 400 hedge trimmers to cover its costs, illustrating the importance of linear relationships in break-even analysis.