Problem 26 - 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

Smallest revenue: \( \$0 \). Largest revenue: \( \$26000 \).

Step 1: Determine Revenue Function

Firstly, we need to establish the revenue based on the number of chairs, \( n \), ordered by the customer. The revenue will be the number of chairs times the price per chair. For orders up to 300 chairs, the revenue is straightforward: \( R = 90n \). Beyond 300 chairs, the price per chair decreases by \( 0.25(n-300) \). Thus, for \( n > 300 \), the revenue function becomes \( R = (90 - 0.25(n-300))n \).

Step 2: Revenue Expression Simplification

Simplifying the revenue expression for \( n > 300 \), \( R = \left(90 - 0.25(n - 300)\right)n \) becomes \[ R = (90n - 0.25n^2 + 75n). \] Therefore, \( R = 165n - 0.25n^2 \) for \( n > 300 \).

Step 3: Evaluate Smallest Revenue

For the smallest revenue, consider the smallest order size. Evaluate the revenue function at the minimum order, which is 0 chairs: \( R = 90 \times 0 = 0 \). Hence, the smallest revenue is \( \$0 \).

Step 4: Evaluate Largest Revenue

To find the largest revenue, use the expression for \( n > 300 \). Substitute \( n = 400 \) into \( R = 165n - 0.25n^2 \): \[ R = 165 \times 400 - 0.25 \times 400^2. \] Simplifying gives \[ R = 66000 - 40000 = 26000. \] Hence, the largest revenue possible is \( \$26000 \).

Key Concepts

Piecewise Functions

In mathematics, piecewise functions allow us to describe a situation where a function is defined by different expressions based on different intervals of the input variable. In the context of the furniture business, the revenue function is piecewise. This means it's defined by two different rules, depending on the number of chairs ordered.

- For orders up to 300 chairs, the revenue function is simply: \( R = 90n \). Here, each chair costs $90, thus the revenue increases linearly with each chair purchased.
- For orders exceeding 300 chairs, the revenue function adapts to a new formula: \( R = (90 - 0.25(n-300))n \). This formula accounts for a price reduction beyond 300 chairs, illustrating the piecewise nature of the function.
The ability to express such variable-dependent outcomes through piecewise functions is crucial for accurately modeling and understanding real-world scenarios where conditions change with quantity or time.

Pricing Strategies

Pricing strategies in business are crucial for maximizing revenue and remaining competitive. In this scenario, the pricing strategy hinges on offering a discounted rate beyond a certain quantity of chairs. This strategy can help encourage larger orders by making it more economically appealing to the customer.

- **Fixed Pricing**: Up to 300 chairs, the price is fixed at \(90 per chair. This simplifies cost calculation for the customer and ensures a steady revenue rate for each chair.
- **Discount Approach**: Beyond 300 chairs, the price per chair decreases by \)0.25 for each additional chair purchased. For instance, if 310 chairs are ordered, the price per chair deviates from $90, becoming: \( 90 - 0.25 imes 10 = 87.5 \). This markdown can incentivize larger orders, aiming to increase volume and total revenue.
A smart pricing strategy matches customer expectations with business objectives, balancing between competitive pricing and profitability.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, often represented in the standard form as \( ax^2 + bx + c \). They are characterized by their parabolic graphs, which can open upwards or downwards depending on the coefficient of the quadratic term. In this problem, the revenue function for orders over 300 chairs is quadratic:
\[ R = 165n - 0.25n^2 \].This equation represents a downward-opening parabola since the coefficient of \( n^2 \) (which is \(-0.25\)) is negative. Quadratic functions can be maximized or minimized subject to their vertex, a critical feature used to determine the optimal point, in this case, the revenue point.
By evaluating this function at specific values within its domain, such as 400 chairs in this scenario, one can determine the maximum revenue achievable, which in this case is \$26,000. However, beyond a certain point, increasing orders may lead to diminishing returns due to decreased prices, a characteristic behavior of quadratic functions.