Problem 32 - Applied Calculus

Question

A company has a total budget of $$\$ 500,000$$ and spends this budget on raw materials and personnel. The company uses \(m\) units of raw materials, at a cost of $$\$ 100$$ per unit, and hires \(r\) employees, at a cost of $$\$ 25,000$$ each. (a) What is the equation of the company's budget constraint? (b) Solve for \(m\) as a function of \(r\). (c) Solve for \(r\) as a function of \(m\).

Step-by-Step Solution

Answer

(a) The equation is \( 100m + 25000r = 500000 \). (b) \( m = 5000 - 250r \). (c) \( r = 20 - \frac{m}{250} \).

Step 1: Establish the Budget Constraint Equation

To establish the budget constraint, consider the costs incurred from raw materials and personnel. The cost for raw materials is calculated as \( 100m \) dollars (since each unit costs \(100), and the cost for personnel is \( 25000r \) dollars (since each employee costs \)25,000). Both these costs combined should equal the total budget of $500,000. Thus, the budget constraint equation is given by \( 100m + 25000r = 500000 \).

Step 2: Solve for m as a Function of r

To express \( m \) in terms of \( r \), we rearrange the budget constraint equation. Starting with \( 100m + 25000r = 500000 \), subtract \( 25000r \) from both sides to get \( 100m = 500000 - 25000r \). Then, divide both sides by 100 to isolate \( m \). The equation becomes \( m = \frac{500000 - 25000r}{100} \). Simplifying, we find \( m = 5000 - 250r \).

Step 3: Solve for r as a Function of m

Similarly, to solve for \( r \) as a function of \( m \), we start with the same budget equation: \( 100m + 25000r = 500000 \). Subtract \( 100m \) from both sides to obtain \( 25000r = 500000 - 100m \). Next, divide both sides by 25000 to isolate \( r \), resulting in \( r = \frac{500000 - 100m}{25000} \). Simplifying, the equation becomes \( r = 20 - \frac{m}{250} \).

Key Concepts

Budget Allocation

In any organization, budget allocation is the process of distributing financial resources across different areas to ensure that all operations run smoothly. It involves deciding how much money to allocate to different needs such as raw materials, salaries, infrastructure, and more. This particular exercise focuses on splitting a total budget between raw materials and personnel. The company has a budget of $500,000 that must be carefully managed. To determine the distribution:

• The cost of raw materials is $100 per unit.
• The cost for hiring each employee is $25,000.

When these costs are combined to form the budget constraint, we ensure that every dollar is accounted for and properly allocated based on immediate business priorities. Effective budget allocation helps to optimize resource use, making it critical in strategic planning.

Equation Solving

Equation solving comes into play when determining how to manage and adjust expenses to stay within a defined budget. In this scenario, given that the total budget is $500,000, you need to balance it against the costs of materials and employees.By setting up an equation, you can define the relationship between these costs. The budget constraint equation, representing both raw materials and personnel, captures this balance:\[100m + 25000r = 500000\]To solve for either variable, you isolate one side to express each resource (raw materials or personnel) in terms of the other. Consider solving for the number of raw materials:1. Start with the equation: \[100m = 500000 - 25000r\]2. Divide by 100 to solve for \(m\):\[m = \frac{500000 - 25000r}{100} = 5000 - 250r\]This process of rearranging and solving equations allows you to understand the trade-offs involved in budget decisions.

Resource Management

Resource management involves strategically managing company assets to maximize operational efficiency. It is crucial for businesses aiming to optimize productivity while minimizing costs. By managing resources wisely, companies can prevent overspending and make the most out of their budget.In this exercise, managing resources involves balancing the use of raw materials and hiring the optimal number of employees. The challenge lies in allocating resources without exceeding the budget.To find the right number of employees (\(r\)) based on the raw materials (\(m\)) used, you solve the constraint equation in reverse:1. Begin with the budget equation: \[25000r = 500000 - 100m\]2. Divide by 25,000 to solve for \(r\): \[r = 20 - \frac{m}{250}\]This not only illustrates resource allocation but also demonstrates how organizations manage constraints such as a fixed budget, to align their resources with operational goals effectively.