Problem 13 - Applied Calculus

Question

The quantity, \(Q,\) of a good produced depends on the quantities \(x_{1}\) and \(x_{2}\) of two raw materials used: $$ Q=x_{1}^{0.3} x_{2}^{0.7} $$ A unit of \(x_{1}\) costs \(\$ 10,\) and a unit of \(x_{2}\) costs \(\$ 25 .\) We want to maximize production with a budget of \(\$ 50\) thousand for raw materials. (a) What is the objective function? (b) What is the constraint?

Step-by-Step Solution

Answer

Objective: Maximize \(Q(x_1, x_2) = x_1^{0.3} x_2^{0.7}\). Constraint: \(10x_1 + 25x_2 \leq 50000\).

Step 1: Understand the Objective

We aim to maximize production, which is represented by the quantity function \( Q = x_1^{0.3} x_2^{0.7} \). The objective function, therefore, is the production function that we wish to maximize.

Step 2: Determine the Objective Function

The objective function is simply the production function itself. We write:\[ Q(x_1, x_2) = x_1^{0.3} x_2^{0.7} \] This is the function we need to maximize given the budget constraint.

Step 3: Define the Cost Function

Each unit of \(x_1\) costs \\(10 and each unit of \(x_2\) costs \\)25. We need to formulate the budget constraint using these cost values.

Step 4: Establish the Budget Constraint

The total cost of the materials is given by:\[ 10x_1 + 25x_2 \leq 50000 \] This inequality represents the budget constraint, considering the budget is \$50,000.

Step 5: Express the Constraint Explicitly

To simplify, express the constraint explicitly:The constraint is: \[ 10x_1 + 25x_2 = 50000 \] Under the budget limit, where the maximum budget should not be exceeded.

Key Concepts

Objective Function

In optimization problems, the objective function is a crucial concept. It represents the quantity or goal we want to optimize, meaning either maximize or minimize. Here, the goal is to maximize the production output, denoted by \( Q \). This output is a function of the two raw materials, \( x_1 \) and \( x_2 \).

The objective function given in the exercise is defined as:
\[ Q(x_1, x_2) = x_1^{0.3} x_2^{0.7} \]

This means the production quantity \( Q \) is achieved by combining the two raw materials in a specific manner. Each raw material contributes to the production differently, as shown by the powers of 0.3 and 0.7, representing the elasticity or responsiveness of output to the inputs. Maximizing this function under certain constraints will give us the ideal production setup.

Budget Constraint

Constraints are limitations or conditions imposed on an optimization problem. A budget constraint is a common type of constraint where the total cost of resources must not exceed a pre-determined limit. In this exercise, the budget constraint is based on the cost of the raw materials \( x_1 \) and \( x_2 \).

Each unit of \( x_1 \) costs \(10 and each unit of \( x_2 \) costs \)25. We are given a budget of \(50,000 to spend on these materials.

The constraint equation can be expressed as:
\[ 10x_1 + 25x_2 \leq 50000 \]

This equation ensures that the spending on raw materials does not exceed \)50,000. Keeping within this budget while maximizing production is key to solving this problem.

Production Maximization

Production maximization is the primary goal in this exercise. It involves adjusting the proportions of \( x_1 \) and \( x_2 \) to achieve the highest possible output \( Q \) given the budget constraint.

- The powers in the objective function \( x_1^{0.3} \) and \( x_2^{0.7} \) suggest how much each material impacts the production level.
- The goal is to find the optimal combination of \( x_1 \) and \( x_2 \) that allows for the maximum value of \( Q \) without exceeding the budget.

Understanding elasticity helps in decision-making here, as it tells you how responsive the output is to changes in input quantities. By maximizing production, businesses efficiently utilize resources, ensuring growth and profitability.

Raw Materials Cost

The cost of raw materials is an essential factor impacting both the budget constraint and the production maximization objective. It directly influences how much of each material can be used.

- Each unit of \( x_1 \) costs \(10.
- Each unit of \( x_2 \) costs \)25.

Given these costs, the way funds are allocated between \( x_1 \) and \( x_2 \) significantly affects production. The challenge lies in optimizing material purchases within the budget limit to achieve the best output. Devising a strategy that accounts for both cost and necessary material proportions is pivotal to solving the problem effectively.