Problem 43 - College Algebra and Calculus: An Applied Approach

Question

Optimal Revenue An accounting firm charges $$\$ 2500$$ for an audit and $$\$ 350$$ for a tax return. Research and available resources have indicated the following constraints. \- The firm has 900 hours of staff time available each week. \- The firm has 155 hours of review time available each week. \- Each audit requires 75 hours of staff time and 10 hours of review time. \- Each tax return requires \(12.5\) hours of staff time and \(2.5\) hours of review time. What numbers of audits and tax returns will bring in an optimal revenue?

Step-by-Step Solution

Answer

The optimal solution will be the coordinates of the vertex that maximizes the revenue function.

Step 1: Define the Variables

Let \( x \) represent the number of audits and \( y \) represent the number of tax returns.

Step 2: Formulate the Objective Function

The total revenue that the firm gets can be represented as \( R = 2500x + 350y \)

Step 3: Formulate the Constraints

The constraints from staff time availability are \( 75x + 12.5y \leq 900 \) and review time availability constraints are \( 10x + 2.5y \leq 155 \)

Step 4: Solve the Problem

Solving the system of inequalities, the constraint region is found. This region is plotted on a graph. The vertices of the plot region are calculated, as the maximum or minimum values of the function occur at these vertices. The coordinates (number of audits and tax returns) that maximize the revenue function is the solution.

Key Concepts

Objective Function

In linear programming, the objective function is the formula we want to optimize. In the given problem, the objective is to maximize revenue. Each audit generates \(2500, and each tax return brings \)350. So, the revenue R can be represented mathematically as \( R = 2500x + 350y \). Here, \( x \) stands for the number of audits, and \( y \) for the number of tax returns.
Understanding the objective function is crucial because it tells us what result we are aiming for. It's the driving force of the problem, aiming to give us either the highest or lowest possible outcome, depending on whether we wish to maximize or minimize. In our case, it's all about maximizing the firm's revenue.

Constraints

Constraints are the limitations or restrictions we need to consider. In our exercise, the accounting firm faces restrictions on time.

• For staff time: An audit takes 75 hours and a tax return takes 12.5 hours. The firm has 900 hours available, leading to the constraint \( 75x + 12.5y \leq 900 \).
• For review time: An audit requires 10 hours, and a tax return requires 2.5 hours. With a total of 155 hours available, the constraint is \( 10x + 2.5y \leq 155 \).

These inequalities outline the conditions under which the objective function is calculated. Constraints are essential as they reflect real-world limitations, ensuring solutions are feasible.

System of Inequalities

When constraints are expressed mathematically, they form a system of inequalities. In simple terms, it is a collection of multiple inequalities that define the problem's feasible region.
For this accounting firm, there are two main inequalities:

• \( 75x + 12.5y \leq 900 \)
• \( 10x + 2.5y \leq 155 \)

These inequalities together describe the boundary within which we search for the solution. By graphing these inequalities, an intersection or overlap of feasible areas appears, which is crucial for determining optimal solutions.
The "feasible region" formed by these inequalities is important because it dictates where the solution can be found. Solutions outside this area violate the constraints.

Optimal Revenue

The goal of linear programming in this context is obtaining optimal revenue, which means finding the highest possible revenue given the constraints. This involves examining the vertices of the feasible region.
Vertices are points where two or more boundary lines intersect in the graph of a system of inequalities. These are crucial because, in linear programming problems, maximum or minimum values of the objective function occur at these vertices.
For the accounting firm, after plotting the constraints on a graph, the intersection points—or vertices—are examined. By evaluating the objective function at these vertices, the combination of audits and tax returns that provides the maximum revenue can be found. This process ensures that the firm's resources are utilized most efficiently, under the given conditions.