Problem 70 - College Algebra and Calculus: An Applied Approach

Question

Break-Even Analysis You invest \(\$ 3000\) in a fishing lure business. A lure costs \(\$ 1.06\) to produce and will be sold for \(\$ 5.86 .\) How many lures must you sell to break even?

Step-by-Step Solution

Answer

To break-even, approximately 625 lures must be sold.

Step 1: Identify the Costs

The total cost for the business is the sum of the initial investment and the cost to produce each fishing lure. The initial investment is \(\$3000\) and it costs \(\$1.06\) to produce each lure. If we let \(x\) be the number of lures produced, then the total cost can be represented as \(3000 + 1.06x\).

Step 2: Identify the Revenue

The total revenue for the business comes from selling each fishing lure at a given price. In this case, each lure is sold for \(\$5.86\). So, if \(x\) is the number of lures sold, then the total revenue can be represented as \(5.86x\).

Step 3: Set up the Equation

The break-even point occurs when the total cost is equal to the total revenue. Therefore, we can equate our expressions for total cost and total revenue to obtain the equation \(3000 + 1.06x = 5.86x\).

Step 4: Solve the Equation

To solve for \(x\), we subtract \(1.06x\) from both sides to get \(3000 = 4.8x\). Finally, we divide both sides by \(4.8\) to isolate \(x\). Thus, \(x = 3000 / 4.8 \approx 625\).

Key Concepts

Cost Analysis

Cost analysis is a crucial part of any business strategy, as it helps entrepreneurs understand the expenses involved before starting or running a business. In the case of the fishing lure business, it is essential to consider both fixed and variable costs to determine the total expenditure.

The fixed cost for our business example is the initial investment of \(3000, which is needed to start production. This cost remains constant regardless of how many lures are produced. It could include expenses such as lease for premises, initial equipment purchase, or other setup costs.

The variable cost, on the other hand, depends on the number of products being manufactured. Each fishing lure costs \)1.06 to produce, and this cost varies directly with production. Therefore, the variable cost is calculated by multiplying $1.06 by the number of lures produced, symbolized as \(1.06x\), where \(x\) denotes the number of lures.

Adding both fixed and variable costs gives the total costs as \(3000 + 1.06x\). Understanding these costs helps in setting realistic financial goals and determining the pricing strategy.

Revenue Calculation

Revenue calculation is an integral part of understanding how well a business can potentially perform. Revenue is the income generated from selling goods or services, which, in turn, should cover costs and bring profit.

For our fishing lure business, revenue is generated from selling lures, where each is priced at $5.86. Therefore, the revenue depends on the volume of sales.

To compute the revenue, one would multiply the price per lure by the total number of lures sold, expressed as \(5.86x\). Here, \(x\) is the number of units sold. This simple multiplication gives the total revenue, as it provides the overall income from all units sold and helps assess whether the business can cover its costs and potentially earn profits.

• Revenue Equation: \(5.86x\)

This calculation is vital to determine whether the revenue can meet or exceed the total costs calculated from the cost analysis, particularly when approaching break-even.

Algebraic Equations

Algebraic equations provide the mathematical framework to solve real-world problems like break-even analysis. In this context, an equation is set up to find out how many units need to be sold for costs to equal revenue.

To compute the break-even point for our fishing lure business, we established the equation \(3000 + 1.06x = 5.86x\). This equation balances the total costs with total revenue to find where they are equal, meaning no loss or profit is made.

The solution involves rearranging this equation to solve for \(x\), representing the number of units that need to be sold to break even. In our example:

• First, subtract \(1.06x\) from both sides: \(3000 = 4.8x\)
• Then, divide each side by 4.8 to isolate \(x\): \(x = \frac{3000}{4.8}\)

This computation shows \(~625\) units need to be sold. Utilizing algebra helps efficiently resolve such problems, fostering informed business decisions.