Problem 34 - Intermediate Algebra

Question

In Problems \(34-37\) recall that the money a business spends to produce a product (or service) is called its cost and the money that it takes in from the sales of a product (or service) is called the revenue. In business and economics, it is important to determine the value at which costs equal the revenue, called the break-even point. Cosmetology. \(\quad\) A beauty shop specializing in hair color treatments has fixed costs of \(\$ 2,310\) per month. The owner estimates that the cost for each treatment is \(\$ 22,\) which covers labor, chemicals, and electricity. a. If her shop can give as many hair color treatments as she wants at a price of \(\$ 64\) each, how many must be given each month to break even? (Hint: To break even, revenue \(=\) costs.) b. How many hair color treatments must be given each month to make a profit?

Step-by-Step Solution

Answer

To break even, 55 treatments are needed; to make a profit, more than 55 treatments are required.

Step 1: Understand Cost Function

The shop's cost includes fixed costs and variable costs. The fixed cost is \( 2310 \) dollars per month. The variable cost for each treatment is \( 22 \) dollars. The cost function can be expressed as \( C(x) = 2310 + 22x \), where \( x \) is the number of treatments.

Step 2: Understand Revenue Function

The revenue generated from selling \( x \) treatments at \( 64 \) dollars each is given by the revenue function \( R(x) = 64x \).

Step 3: Set Up the Equation to Find Break-even Point

The break-even point occurs where costs equal revenue: \( R(x) = C(x) \). This means \( 64x = 2310 + 22x \).

Step 4: Solve for x in Break-even Equation

Rearrange \( 64x = 2310 + 22x \) to isolate \( x \): \[64x - 22x = 2310 \] This simplifies to \[42x = 2310 \]Divide both sides by \( 42 \): \[x = \frac{2310}{42} = 55 \]. So, 55 treatments are needed to break even.

Step 5: Calculate Minimum for Profit

To make a profit, the revenue must be greater than the costs. Therefore, solve \( 64x > 2310 + 22x \). Rearrange it: \[64x - 22x > 2310 \] This simplifies to \[42x > 2310 \]Divide both sides by \( 42 \): \[x > \frac{2310}{42} = 55 \].Therefore, more than 55 treatments are needed to make a profit.

Key Concepts

Cost Function

In the context of a beauty shop, the cost function is crucial for understanding the expenses incurred by the business. A cost function is a mathematical expression that captures both fixed and variable costs.
Fixed costs are those that remain constant regardless of the number of treatments given. For the beauty shop, these are the costs that need to be paid each month, like rent or leasing costs, amounting to \(2310\) dollars.
Variable costs depend on the number of units produced—in this case, hair color treatments. Here, this cost is \(22\) dollars per treatment, which covers labor, chemicals, and electricity.
The cost function, therefore, can be expressed mathematically as:

• \(C(x) = 2310 + 22x\), where \(x\) represents the number of treatments performed.

This equation allows the business to calculate total costs at varying levels of activity.

Revenue Function

The revenue function explains how much money the business makes from selling its services. It's a vital calculation to ensure that earnings can cover costs, and ideally, generate profit.
For the beauty shop, each hair color treatment is sold for \(64\) dollars. Therefore, the revenue that comes from selling \(x\) treatments can be calculated by:

• \(R(x) = 64x\)

This function is straightforward: simply multiply the price per service by the number of services rendered.
The revenue function will help the shop owner determine how many treatments need to be sold to meet the costs and start making a profit.

Fixed Costs

Fixed costs are a fundamental part of the cost structure in any business, including the beauty shop. These are the expenses that don't change with the level of production or sales; they remain steady regardless of the number of treatments performed.

• Examples of fixed costs include rent, insurance, salaries of permanent staff, and equipment leasing.

In this scenario, the beauty shop has a fixed monthly cost of \(2310\) dollars.
Understanding fixed costs is crucial for planning and ensuring the business at least covers these base expenses each month.
These costs must be accounted for even when calculating the break-even point or when strategizing for profit.

Variable Costs

Variable costs refer to the expenses that fluctuate in direct proportion to the level of production or sales volume. Unlike fixed costs, they change with the number of hair color treatments provided.

• For each treatment, the beauty shop incurs \(22\) dollars in costs, which cover the necessary inputs such as labor, chemicals, and electricity.

Thus, variable costs directly impact the total cost, based on how many services are offered.
Monitoring variable costs is crucial, especially when estimating how many treatments need to be sold to reach a break-even point.
By fully understanding variable costs, the business can make informed decisions on pricing strategies and cost management to maximize profits.