Problem 27 - Applied Calculus

Question

You run a small furniture business. You sign a deal with a customer to deliver up to 400 chairs, the exact number to be determined by the customer later. The price will be \(\$ 90\) per chair up to 300 chairs, and above 300 , the price will be reduced by \(\$ 0.25\) per chair (on the whole order) for every additional chair over 300 ordered. What are the largest and smallest revenues your company can make under this deal?

Step-by-Step Solution

Answer

Largest revenue: $27,000; Smallest revenue: $90.

Step 1: Understand the Pricing Structure

First, note that when the customer orders up to 300 chairs, the price per chair is constant at $90. For orders above 300 chairs, the price per chair decreases by $0.25 for each chair above 300.

Step 2: Identify Revenue for Up to 300 Chairs

For orders between 1 and 300 chairs, the revenue is calculated as \( R = 90n \) where \( n \) is the number of chairs ordered (\(1 \leq n \leq 300\)).

Step 3: Identify Revenue for More Than 300 Chairs

For orders above 300 chairs, the revenue formula changes. The price per chair becomes \( P = 90 - 0.25(k - 300) \) if \( k \) is more than 300 chairs. Then the revenue \( R \) is given by \( R = k \times P = k \times (90 - 0.25(k - 300)) \).

Step 4: Calculate Revenue for 400 Chairs

Calculate the revenue for the maximum order of 400 chairs. Here, the price reduction is due to 100 additional chairs over 300, so \( P = 90 - 0.25 \times 100 = 65 \). The revenue is \( R = 400 \times 65 = 26000 \).

Step 5: Calculate Maximum Revenue for 300 Chairs

For exactly 300 chairs, the revenue is calculated as \( R = 300 \times 90 = 27000 \).

Step 6: Determine Largest and Smallest Revenue

The smallest revenue possible is if the customer orders 1 chair, which earns \( 90 \times 1 = 90 \). The largest revenue is calculated as the maximum for up to 300 chairs, which is 27000 for 300 chairs.

Key Concepts

Revenue Calculation

When operating a business, understanding how to calculate revenue is essential. Revenue is the total income a company generates by selling its goods or services. In this problem, our business model revolves around selling chairs, where the revenue structure changes depending on the number of chairs sold.

• Basic Formula for Revenue: The basic formula for revenue is \[ R = ext{price per unit} \times ext{number of units sold} \] In our case, for up to 300 chairs, this is simply \( R = 90n \), where \( n \) represents the number of chairs.
• Variable Pricing Above 300 Chairs: Beyond 300 chairs, each extra chair sees a reduction in price, impacting the overall revenue calculation. The adjusted price needs to be computed first before calculating revenue.
• Calculating Revenue for Different Quantities: This involves substituting the number of units (chairs) in the formula to compute the total revenue under various scenarios.

In any business setting, understanding these basic calculations helps in financial forecasting and strategic decision-making.

Piecewise Function

A piecewise function is a function composed of multiple sub-functions, each of which applies to a certain interval of the main function's domain. For our furniture business, the pricing structure creates a perfect situation for using a piecewise function.

• Price up to 300 Chairs: The price remains constant at \(\\(90\) per chair, making the revenue function for this segment simple and direct.
• Price for More than 300 Chairs: For quantities more than 300, the price drops by \(\\)0.25\) per chair for each additional chair. This requires a modification to the original revenue function using a different formula for the piece above 300.

This reflects how businesses often have thresholds at which they apply different rules or discounts, and capturing this in a piecewise format aids in accurately modeling such scenarios.

Price Structure Analysis

Analyzing the price structure is crucial to understanding how revenue is impacted under different conditions. Price structures are designed according to customer demand, cost efficiency, and competitive dynamics.

• Fixed Pricing Model for Low Quantities: A fixed price of \(\$90\) per chair applies to orders from 1 to 300 chairs. This simplicity helps in setting a baseline expectation for customers.
• Discounted Pricing Model for High Quantities: Prices decrease for orders exceeding 300 chairs. This structure incentivizes customers to buy more by reducing the cost per chair after a certain threshold.
• Impact on Revenue: The progressive price reduction beyond 300 units showcases how a strategic discount can lead to different revenue outcomes, depending on order volume.

Such analyses allow companies to tailor offerings that are appealing to customers while maximizing revenue potentials.

Business Mathematics

Business mathematics involves the application of mathematical methods and techniques to solve business-related problems. In our scenario, mathematics is key to optimizing pricing and predicting revenue.

• Importance of Accurate Calculations: Ensuring all prices and revenues are calculated correctly prevents financial loss and assists in profitable decision-making.
• Setting Price Strategies Using Mathematics: Calculating the effects of pricing changes is crucial for setting competitive and attractive prices that drive sales.
• Profit Analysis: Beyond revenue, businesses must analyze costs and profit margins to assess overall viability of pricing schemes.

Understanding these concepts helps businesses apply mathematically sound strategies, improving efficiency and profitability over time.