Problem 60 - College Algebra

Question

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

Each guitar should be sold for $409 to break even. Revenue function: \(R(x) = 409x\).

Step 1: Define the Break-even Point

The break-even point is where the cost of production equals the total revenue. In this case, you sell 150 units. The cost associated with producing 150 guitars is given by the cost function \(C(x)\).

Step 2: Calculate the Cost for 150 Units

To find the cost for producing 150 units, substitute \(x = 150\) into the cost function: \(C(150) = 75 \times 150 + 50,000\).

Step 3: Evaluate the Cost Function

Multiply and add to find the total cost: \(C(150) = 75 \times 150 + 50,000 = 11,250 + 50,000 = 61,250\).

Step 4: Write the Revenue Function

The revenue from selling \(x\) units of guitars at price \(p\) is expressed as \(R(x) = p \times x\).

Step 5: Set Up the Equation to Find Selling Price

To break even, the revenue should equal the cost: \(R(150) = C(150)\). Substitute to get \(150p = 61,250\).

Step 6: Solve for the Price per Guitar

Solve the equation \(150p = 61,250\) to find \(p\). Divide both sides by 150 yielding \(p = \frac{61,250}{150}\).

Step 7: Calculate the Exact Price

Perform the division: \(p = \frac{61,250}{150} \approx 408.33\).

Step 8: Round the Price

Since the price should be rounded up to the nearest dollar, the price per guitar is \(409\) dollars.

Key Concepts

Cost Function

The cost function is a critical component in break-even analysis. It represents the total cost incurred by a business to produce a given number of units. For this exercise, the cost function is described by the equation \( C(x) = 75x + 50,000 \).
This equation breaks down into two parts:

• **Variable Costs**: \( 75x \) is the variable cost, which depends on the number of units \( x \) produced. Each unit costs \(75 to make, meaning the total variable cost increases with every additional unit.
• **Fixed Costs**: \)50,000 represents the fixed costs, remaining constant regardless of how many guitars are produced. These are costs the company has to pay, like rent or salaries, that do not change with production volume.

Understanding the cost function helps businesses predict expenses and manage their production costs effectively. Balancing these costs is essential for a company to achieve its desired financial results.

Revenue Function

The revenue function determines the total income from selling a specific number of products at a given price. In this scenario, the revenue function is expressed as \( R(x) = p \times x \). Here, \( p \) is the selling price per unit, and \( x \) represents the quantity sold.

• Revenue is calculated by multiplying the selling price by the number of units sold.
• For break-even analysis, the company needs the revenue to match the production costs at the specified unit level.

In this exercise, to find the break-even point when selling 150 guitars, the revenue function was equated to the cost function at 150 units.
This approach determines the minimum price at which the product must be sold to avoid any losses, ultimately influencing strategic pricing decisions.

Unit Selling Price

The unit selling price is the price at which each individual guitar must be sold. For the guitar factory to reach the break-even point, it must set the selling price such that the total revenue equals the total cost.
Here’s how it is calculated:

• The equation \( 150p = 61,250 \) signifies the total revenue needed to match the cost of producing 150 guitars.
• To isolate \( p \), divide both sides by 150: \( p = \frac{61,250}{150} \).
• Compute the division: \( p \approx 408.33 \).
• Since prices need to be whole numbers, round up to $409.

This calculated price ensures that the revenue from selling each guitar precisely covers both variable and fixed production costs when 150 units are sold.

Cost of Production

The cost of production includes all expenses incurred in producing a certain amount of goods. These costs can be categorized mainly into fixed and variable costs.
In the case of the guitar factory, the cost function \( C(x) = 75x + 50,000 \) captures both:

• **Variable Costs**: At \(75 per guitar, these costs vary directly with the number of guitars produced. For every additional guitar, the total production cost rises by \)75.
• **Fixed Costs**: Totaling $50,000, they remain unchanged no matter how many guitars are manufactured. Examples include overhead costs like administration or lease payments.

Understanding these costs is critical for strategic financial planning. The company needs to accurately estimate what it costs to produce their goods to set appropriate sales prices and achieve profitability.