Problem 35 - 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) Budget constraint: \(100m + 25,000r = 500,000\). (b) \(m = 5,000 - 250r\). (c) \(r = 20 - 0.004m\).

Step 1: Understand the Problem

The company has a budget of \(500,000. It spends this budget on raw materials costing \)100 per unit and hiring employees costing $25,000 each. We need to derive the budget constraint equation, express \(m\) in terms of \(r\), and \(r\) in terms of \(m\).

Step 2: Derive the Budget Constraint Equation

The total cost for \(m\) units of raw materials is \(100m\), and the total cost for \(r\) employees is \(25,000r\). These costs must add up to the total budget: \[ 100m + 25,000r = 500,000. \] This equation represents the budget constraint.

Step 3: Solve for \(m\) as a Function of \(r\)

To express \(m\) as a function of \(r\), rearrange the budget constraint equation: \[ 100m = 500,000 - 25,000r. \] Divide both sides by 100 to solve for \(m\): \[ m = \frac{500,000 - 25,000r}{100}. \] Simplify the expression: \[ m = 5,000 - 250r. \] Thus, \(m\) as a function of \(r\) is \(m = 5,000 - 250r\).

Step 4: Solve for \(r\) as a Function of \(m\)

Now, rearrange the equation \(100m + 25,000r = 500,000\) to express \(r\) as a function of \(m\): \[ 25,000r = 500,000 - 100m. \] Divide both sides by 25,000 to solve for \(r\): \[ r = \frac{500,000 - 100m}{25,000}. \] Simplify the expression: \[ r = 20 - 0.004m. \] Thus, \(r\) as a function of \(m\) is \(r = 20 - 0.004m\).

Key Concepts

Linear Equations

Linear equations are equations of a straight line, which means they have no curves. In our budget constraint problem, these represent relationships between two variables: the number of raw materials the company uses, denoted as \(m\), and the number of employees hired, denoted as \(r\). When we write the budget constraint as \[ 100m + 25,000r = 500,000, \]we create a linear equation. This equation shows how spending on materials and employees affects the company's total budget. By rearranging this equation, we can express each variable in terms of the other. This is useful because it allows us to see different combinations of \(m\) and \(r\) that will still respect the same budget limit. Such an approach is crucial in planning and decision-making.

Resource Allocation

Resource allocation involves distributing available resources among various activities or departments. This concept is vital for businesses, like the company in the exercise, to maximize output or efficiency using limited means. In the budget constraint problem \[ 100m + 25,000r = 500,000, \]this allocation problem becomes math-focused. Each decision on how many materials to buy or employees to hire (\(m\) or \(r\)) directly impacts how the company can use its budget effectively.

• The company needs to determine how many units of raw materials \(m\) are needed without overspending on employees \(r\).
• Finding a balance ensures that the company optimizes its budget effectively, avoiding shortages or excesses.
• Decisions are made based on potential returns and costs, making mathematical solutions indispensable.

Understanding and solving these equations helps highlight opportunities and constraints, aiding in strategic resource planning.

Cost Analysis

Cost analysis is the process of understanding the different costs involved in decision-making. It helps businesses evaluate total expenses to decide where cuts can be made or where additional investment may be required. In this exercise, the company has identified two main components of their costs:

• \(100m\) for the cost of raw materials, meaning \(m\) units of material cost \(100 each.
• \(25,000r\) for the cost of hiring employees, where each employee costs \)25,000.

Total costs should not exceed the company's budget of $500,000. By evaluating the costs associated with \(m\) and \(r\), the business can pinpoint areas for cost reduction or expansion.
The underlying calculations provide the simplest way to monitor these expenses against available budget. This process enables strategic shifts without overstepping financial limits. Effective cost analysis thus ensures financial health and the potential for growth.