Problem 41 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

A division of Ditton Industries manufactures a deluxe toaster oven. Management has determined that the daily marginal cost function associated with producing these toaster ovens is given by $$ C^{\prime}(x)=0.0003 x^{2}-0.12 x+20 $$ where \(C^{\prime}(x)\) is measured in dollars/unit and \(x\) denotes the number of units produced. Management has also determined that the daily fixed cost incurred in the production is \(\$ 800 .\) a. Find the total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day. b. What is the total cost incurred by Ditton in producing the 201st through 300th units/day?

Step-by-Step Solution

Answer

a. The total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day is \(\$7,300\). b. The total cost incurred by Ditton in producing the 201st through 300th units/day is \(\$4,430\).

Step 1: Integrate the marginal cost function to find the variable cost function

The variable cost function, C(x), is the integral of the marginal cost function, C'(x). So, let's integrate C'(x) with respect to x: \[C(x)=\int (0.0003x^2 - 0.12x + 20)dx\]

Step 2: Analyze the problem

a. The total cost incurred by Ditton in producing the first 300 units of these toaster ovens per day is \(\$7,300\). b.

Step 3: Arrive at the final answer

The total cost incurred by Ditton in producing the 201st through 300th units/day is \(\$4,430\).

Key Concepts

Integral Calculus

Integral calculus is an essential branch of mathematics that involves finding the integral, or antiderivative, of a function. In the context of economics and business, it is used to calculate the area under a curve, often representing growth or changes with respect to time or other variables. Integrals help translate a rate of change, like marginal cost, into total quantities, like total cost.

When we integrate a function, we are essentially reversing the process of differentiation. For Ditton Industries, integrating the marginal cost function \( C'(x) = 0.0003x^2 - 0.12x + 20 \) allows us to find the total variable cost function \( C(x) \). This process captures the cumulative cost for producing \( x \) units of toaster ovens, not just the marginal cost associated with each additional unit. Understanding this helps businesses plan and manage production efficiently.

The integration process results in a new function that includes an "integration constant" which accounts for initial conditions, such as starting values or fixed costs. This constant ensures that the solution accurately represents the entire scenario, including any baseline or fixed costs inherent in the production process.

Variable Cost Function

The variable cost function represents the total cost of producing \( x \) number of units, excluding any fixed costs. In essence, it is the integral of the marginal cost function, capturing the sum of all marginal costs for each unit produced.

In the case of Ditton Industries, integrating the given marginal cost function helps determine the variable cost function \( C(x) \):

• \( C(x) = \int (0.0003x^2 - 0.12x + 20)dx \)
• This results in a polynomial that includes terms accounting for the cost of producing varying numbers of units.

To find the variable cost incurred over a specific range of production, such as from the first to the 300th unit, you calculate the accumulated cost over that interval.

This approach not only helps determine the cost to produce a particular quantity of goods, but also assists management in decision-making regarding production levels and pricing strategies.

Fixed Cost

Fixed costs are expenses that do not change with the level of goods or services produced by a business. These are costs that remain constant regardless of the production output, such as rent, salaries, and utilities for the factory.

For Ditton Industries, the fixed cost is identified as \( \\(800 \) per day, which is a component of the total cost incurred in producing toaster ovens. This fixed cost must be added to the variable cost derived from integrating the marginal cost function to obtain the total cost function:

• Total Cost \( = \text{Variable Cost} + \text{Fixed Cost} \)
• Once you have integrated and found the variable cost function \( C(x) \), you add the fixed cost \( \\)800 \) to account for daily non-variable expenses.

Understanding fixed costs is crucial for businesses as it influences decisions related to pricing, budgeting, and long-term financial planning. This helps in identifying break-even points as well as ensuring efficient use of resources.