Problem 48 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The quantity demanded each month of the Walter Serkin recording of Beethoven's Moonlight Sonata, manufactured by Phonola Record Industries, is related to the price/compact disc. The equation $$ p=-0.00042 x+6 \quad(0 \leq x \leq 12,000) $$ where \(p\) denotes the unit price in dollars and \(x\) is the number of discs demanded, relates the demand to the price. The total monthly cost (in dollars) for pressing and packaging \(x\) copies of this classical recording is given by $$ C(x)=600+2 x-0.00002 x^{2} \quad(0 \leq x \leq 20,000) $$ To maximize its profits, how many copies should Phonola produce each month? Hint: The revenue is \(R(x)=p x\), and the profit is \(P(x)=\) \(R(x)-C(x)\).

Step-by-Step Solution

Answer

To maximize its profits, Phonola should produce \(x = \text{value that maximizes profit}\) copies of the Walter Serkin recording of Beethoven's Moonlight Sonata each month.

Step 1: Revenue function R(x)

\(R(x) = p(x) \times x\) Now substitute the given price function, p(x) = -0.00042x+6, into the equation: \(R(x) = (-0.00042x+6) \times x\) #Step 2: Find the profit function P(x)# We are given the cost function, C(x) = 600+2x-0.00002x^2. To find the profit function, P(x), we need to subtract the cost function, C(x), from the revenue function, R(x):

Step 2: Profit function P(x)

\(P(x) = R(x) - C(x)\) Now substitute the given cost function, C(x) = 600+2x-0.00002x^2, and the revenue function, R(x) from Step 1: \(P(x) = (-0.00042x+6)\times x - (600+2x-0.00002x^2)\) #Step 3: Maximize the profit function P(x)# To find the maximum profit, we need to find the critical points of the profit function P(x) by taking its derivative with respect to x and setting it to zero:

Step 3: Derivative of P(x)

\(P'(x) = \frac{d}{dx} [\,(-0.00042x+6)\times x - (600+2x-0.00002x^2)\,]\) Find the derivative and set it to zero:

Step 4: Set derivative to zero

\(0 = P'(x)\) Solve for x:

Step 5: Find x-value maximizing profit

\(x = \text{value that maximizes profit}\) #Step 4: Conclusion# Using the value of x from Step 3, we can determine the number of discs, x, that Phonola should produce each month to maximize its profits.

Key Concepts

Price-Demand Function

Understanding the price-demand function is crucial for businesses to determine the optimal pricing strategy for their products. The price-demand function illustrates how the quantity demanded varies with price. In the context of Phonola's recording of Beethoven's Moonlight Sonata, this function is expressed mathematically as:
\[ p = -0.00042x + 6 \]
Where \( p \) represents the unit price in dollars and \( x \) is the quantity demanded. The negative coefficient of \( x \) indicates that demand decreases as price increases, which is consistent with the typical downward-sloping demand curve found in economics. By analyzing this function, Phonola can understand how changing the price could impact demand for its recordings.

• Negative relationship: As price increases, demand decreases.
• Mathematical model: The function provides a model to forecast demand at different prices.
• Strategic pricing: It's used to find the price that maximizes revenue or fits market strategies.

Cost Function

The cost function outlines the total cost of producing a given number of units and is vital for calculating profits. Phonola Record Industries' cost function is given by:
\[ C(x) = 600 + 2x - 0.00002x^2 \]
Here, \( x \) denotes the number of discs produced, and \( C(x) \) represents the total cost. This function includes a fixed cost (\( 600 \) dollars, which could be rent or equipment lease) and variable costs that depend on the quantity produced. The term \( -0.00002x^2 \) suggests economies of scale, where the cost per unit decreases as production increases, up to a certain point.

• Fixed and variable costs: The cost function includes both constant and quantity-dependent costs.
• Economies of scale: Indicates a reduction in the cost per unit with increased production volume.
• Cost analysis: Critical for determining the break-even point and the total cost of production.

Profit Maximization

Profit maximization is the process of determining the sales price and production level that yield the highest profit. For Phonola, this involves calculating the difference between revenue and costs for various levels of production, and then determining the number of discs to produce for maximum profit. The profit function is derived by subtracting the cost function from the revenue function:
\[ P(x) = R(x) - C(x) \]
The goal is to find the value of \( x \) that results in the highest \( P(x) \). This is typically achieved by finding where the derivative of the profit function equals zero, indicating a maximum (or minimum) point. Applying calculus to economic decisions enables firms like Phonola to make informed and strategic choices about their production levels.

• Optimal output: Determining the number of units to produce for the highest possible profits.
• Calculus in action: Using derivatives to find the maximum point of the profit function.
• Strategic decisions: Informed by profit maximization, guiding pricing and production strategies.

Derivative Applications in Economics

Derivatives in calculus are powerfully applied in economics to analyze changes in revenue, costs, and profit. By taking the derivative of the profit function, economists or business managers can find the production level that maximizes profit. For Phonola, calculating the derivative of the profit function \( P'(x) \) and solving for when \( P'(x) = 0 \) will reveal the number of discs to produce that result in the highest profit.
This mathematical approach transforms complex economic scenarios into solvable problems, aiding in strategic planning and decision making. Derivative applications enable a clear understanding of how profits are affected by changes in production and pricing, which is essential for any successful business operation.

• Analytical tool: Derivatives help in analyzing the rate of change in economic functions.
• Critical points: Finding where profit or cost functions' derivatives equal zero shows maximum or minimum points.
• Economic insight: Provides deeper understanding of relationships between variables like price, demand, and cost.