Problem 60 - College Algebra

Question

For the following exercises, solve for the desired quantity. A guitar factory has a cost of production \(C(x)=75 x+50,000\) . If the company needs to break even after 150 units sold, at what price should they sell each guitar? Round up to the nearest dollar, and write the revenue function.

Step-by-Step Solution

Answer

Sell each guitar at $409; revenue function is \(R(x) = 409x\).

Step 1: Understand the Problem

We need to determine the selling price for each guitar to achieve a break-even point given that the cost function is \(C(x) = 75x + 50,000\). The break-even means that the total cost equals the total revenue after selling 150 guitars.

Step 2: Define the Revenue Function

The revenue function, \(R(x)\), is defined as the product of the number of units sold, \(x\), and the price per unit, \(p\). Therefore, \(R(x) = p \cdot x\).

Step 3: Set Up the Break-Even Equation

At break-even, total cost equals total revenue. Therefore, we have: \(R(x) = C(x)\). For 150 units sold, this becomes \(150p = 75(150) + 50,000\).

Step 4: Substitute and Solve for Price

Substitute the number of units (150) into the equation to find the price. Calculate: \[150p = (75 \times 150) + 50,000\]This simplifies to:\[150p = 11,250 + 50,000 = 61,250\]Divide both sides by 150:\[p = \frac{61,250}{150} \approx 408.33\]

Step 5: Round up the Price

Since prices should be whole numbers, and to ensure the company breaks even or better, round up to the nearest dollar:\(p = 409\).

Step 6: Write the Revenue Function

Given the computed price per guitar, the revenue function becomes:\(R(x) = 409x\).

Key Concepts

Cost Function

In any business, understanding your costs is crucial to financial stability. A cost function is a mathematical expression that describes how total costs change with variation in the level of activity or output, such as the number of goods produced, termed as units.

For our guitar factory, the cost function is given by the equation: \(C(x) = 75x + 50,000\). Here's what this means:

• \(x\) represents the number of guitars produced and sold.
• 75 is the variable cost per guitar, implying each additional guitar incurs a cost of $75.
• The term 50,000 represents fixed costs, expenses that do not change regardless of the number of guitars produced, such as rent and salaries.

This function helps businesses forecast expenses based on production levels and determines whether scaling up or down is financially viable.

By analyzing the cost function, businesses can make informed decisions and strategies for pricing and production.

Revenue Function

The revenue function is integral to business as it depicts how income is generated via sales. For our exercise, the revenue function is a key player in determining the break-even point, which is when a business neither loses money nor makes a profit.

Revenue, denoted by \(R(x)\), is calculated by multiplying the number of units sold, \(x\), by the selling price per unit, \(p\). Mathematically, this is expressed as \(R(x) = p \cdot x\).

In our specific case for the guitar factory, we derived that the price per guitar to break even should be $409. Thus, the revenue function becomes:

• \(R(x) = 409x\), where \(409\) is the price per guitar.

With this function, the factory can predict the total revenue from the sale of any number of guitars. Understanding revenue through this lens allows businesses to set financial goals and measure their performance.

Algebraic Equations

Algebraic equations are the instruments used to solve various problems involving unknown quantities. In our context, they are utilized to determine the correct selling price to achieve a break-even point.

To find the break-even price in this scenario:

• We set the total cost function equal to the revenue function: \(C(x) = R(x)\).
• For 150 guitars, this translates to: \(150p = 75(150) + 50,000\).
• Here, \(p\) is the unknown price per guitar we need to solve for.
• Simplifying gives: \(150p = 11,250 + 50,000\), further reducing to \(150p = 61,250\).
• Solving for \(p\), we divide by 150 to find: \(p = \frac{61,250}{150} \approx 408.33\).

Through the use of algebraic equations, we could efficiently determine the unknown value necessary for the factory to reach its financial equilibrium.

Rounding Numbers

Rounding numbers is crucial in financial calculations to ensure practicality and accuracy when dealing with currency. Often, financial figures require rounding to the nearest whole number for ease of transactions.

In our guitar factory problem, the calculated price per guitar was \(408.33\). Since prices are typically set in whole dollars, we need to round.

• For bookkeeping and operational ease, it’s important to round up to the nearest dollar.
• Thus, the price was adjusted from \(408.33\) to \(409\) to ensure no loss occurs from fractional pricing.

Rounding up, especially in business, helps safeguard against losses and provides clear and clean pricing for consumers, ensuring the business maintains positive financial standing.