Problem 27 - Calculus and its applications

Question

Minimizing total cost. Each unit of a product can be made on either machine A or machine B. The nature of the machines makes their cost functions differ: Machine A: \(C(x)=10+\frac{x^{2}}{6}\) Machine B: \(C(y)=200+\frac{y^{3}}{9}\). Total cost is given by \(C(x, y)=C(x)+C(y)\). How many units should be made on each machine in order to minimize total costs if \(x+y=10,100\) units are required?

Step-by-Step Solution

Answer

Produce 10,000 units on Machine A and 100 units on Machine B.

Step 1: Define the Problem

We need to minimize the total cost function, given by \( C(x, y) = C(x) + C(y) = 10 + \frac{x^2}{6} + 200 + \frac{y^3}{9} \). We also know that the total units produced is \( x + y = 10,100 \). Our aim is to find \( x \) and \( y \) such that the total cost is minimized.

Step 2: Substitute Constraint into Cost Function

Use the constraint \( x + y = 10,100 \) to express \( y = 10,100 - x \). Substitute this into the cost function to get a function of a single variable: \[ C(x) = 210 + \frac{x^2}{6} + \frac{(10,100-x)^3}{9} \].

Step 3: Differentiate the Cost Function

Find the derivative of the cost function with respect to \( x \): \[ \frac{dC}{dx} = \frac{x}{3} - \frac{3(10,100 - x)^2}{9} \] which simplifies to \[ \frac{dC}{dx} = \frac{x}{3} - \frac{(10,100 - x)^2}{3} \].

Step 4: Set the Derivative to Zero

To find the minimum cost, set the derivative \( \frac{dC}{dx} \) equal to zero: \[ \frac{x}{3} = \frac{(10,100 - x)^2}{3} \]. This simplifies to \( x = (10,100 - x)^2 \).

Step 5: Solve for x

We solve the equation \( x = (10,100 - x)^2 \). By solving this equation via appropriate algebraic manipulation, we find that \( x \approx 10,000 \) units.

Step 6: Calculate y Using the Constraint

Since \( x + y = 10,100 \), then \( y = 10,100 - 10,000 = 100 \) units.

Key Concepts

Cost Minimization

Cost minimization is a fundamental concept in optimization, especially in calculus, where the goal is to determine strategies that result in the least expense. In many real-world scenarios, businesses need to manufacture products while keeping their costs as low as possible. This involves using mathematical techniques to determine the most cost-effective way of producing a given output. In the context of our exercise, cost minimization involves deciding how many units to produce on each machine, A and B, to achieve the total of 10,100 units at the lowest possible cost.

In our exercise, the total cost function is provided, incorporating the cost behaviors of two different machines. To minimize total cost effectively:

• Understand the cost equations distinctly for each machine.
• Apply any existing constraints, such as the total number of units required.
• Substitute constraints into your cost function to reduce variables, simplifying the minimization process.

This approach ensures the decisions made are supported by an optimal use of resources, thanks to calculus-driven insights.

Differentiation

Differentiation is a key technique in calculus used to find the rate at which quantities change. In the task of optimizing functions, differentiation helps identify where costs or benefits are at their maximum or minimum points. In this exercise, to minimize the cost, differentiation assists in finding the precise point where the rate of change of cost with respect to the quantity produced becomes zero.

Let's break it down:

• First, you differentiate the cost function obtained after substituting the constraints, reducing it to a single-variable function.
• Then, find the derivative with respect to that variable, capturing how cost behaves as you adjust production quantities.
• The final step involves setting this derivative equal to zero and solving for the variable to find points where the cost is minimized.
  Doing so helps us pinpoint exact values needed, ensuring production plans are cost-effective and efficient.

This process is central to optimization problems, making differentiation an indispensable calculus tool.

Mathematical Modeling

Mathematical modeling involves creating a mathematical representation of a real-world scenario to analyze and solve problems effectively. In our cost minimization exercise, mathematical modeling comes into play by using equations to represent the costs associated with each machine.

Here's how it works:

• Begin by defining equations that model the cost functions of each machine individually. These should reflect the specific cost dynamics described in the problem statement.
• Introduce any constraints into your model, such as total units, to ensure the model aligns with real-world conditions.
• Translate the entire situation into a unified equation, facilitating analysis through techniques like differentiation, as previously discussed.

This approach provides a structured framework for solving complex problems. It allows for predictions and decisions to be based on solid, quantitative foundations, transforming abstract mathematical concepts into tangible results applicable to everyday scenarios.