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Chapter 7: Problem 41

Optimal Revenue An accounting firm has 780 hours of staff time and 272 hours of reviewing time available each week. The firm charges \(\$ 1600\) for an audit and \(\$ 250\) for a tax return. Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. What numbers of audits and tax returns will yield an optimal revenue? What is the optimal revenue?

Short Answer

Expert verified
This problem requires a linear programming method to solve. The optimal number of audits (x) and tax returns (y) that will yield the optimal revenue, as well as the value of the optimal revenue, can be found by solving the constraints and the objective function.

Step by step solution

01

Define the Variables

Let \(x\) represent the number of audits and \(y\) represent the number of tax returns. These are the variables we need to determine.
02

Formulate the Constraints

Each audit requires 60 hours of staff time and 16 hours of review time. Each tax return requires 10 hours of staff time and 4 hours of review time. The total staff time and review time cannot exceed 780 hours and 272 hours respectively. The constraints can thus be written as: \(60x + 10y \leq 780\), \(16x + 4y \leq 272\)
03

Formulate the Objective Function

Revenue from an audit is \(\$1600\) and from a tax return is \(\$250\). Thus, the total revenue, which is the objective function to be maximized, can be written as: \(Z = 1600x + 250y\)
04

Solve the Problem

This is a linear programming problem which can be solved graphically or by using a method like the simplex method. The solution will give the optimal values of \(x\) and \(y\) which will yield the maximum revenue.
05

Calculate the Optimal Revenue

Once the values of \(x\) and \(y\) have been found, the optimal revenue can be found by substituting these values into the objective function, \(Z = 1600x + 250y\). This will yield the optimal revenue.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Objective Function
In linear programming, the objective function is the equation that you want to maximize or minimize, often associated with costs, profits, or revenues. In the context of the accounting firm's problem, our goal is to maximize revenue. The firm earns \(1600 per audit and \)250 per tax return. Thus, the objective function is formulated as the total revenue equation: \(Z = 1600x + 250y\).
Here, \(x\) represents the number of audits while \(y\) denotes the number of tax returns.
The main aim is to find the combination of \(x\) and \(y\) that makes \(Z\) as large as possible, given the constraints of available resources.
Constraints
Constraints in linear programming define the limits within which we must operate. They represent the resources available and the requirements of each task. For the accounting firm:
  • Each audit requires 60 hours of staff time and 16 hours of review time.
  • Each tax return consumes 10 hours of staff time and 4 hours of review time.
  • Staff time cannot exceed 780 hours weekly, and review time cannot exceed 272 hours weekly.
This leads to the following system of inequalities:
- \(60x + 10y \leq 780\)
- \(16x + 4y \leq 272\)
These inequalities ensure that the total staff and review time used does not surpass what is available.
Optimal Revenue
Optimal revenue is the highest possible income given the firm's constraints. To achieve this, we solve for the values of \(x\) and \(y\) that maximize the objective function \(Z = 1600x + 250y\).
Once the optimal values of \(x\) and \(y\) are found, they are substituted back into \(Z\) to calculate the maximum possible revenue.
This step ensures that the resource constraints are adhered to while achieving the greatest potential return for the firm.
Graphical Method
The graphical method is a straightforward way to solve linear programming problems when there are only two variables. It involves plotting the constraints as lines on a graph.
The feasible region, which is the set of all possible solutions, is the area where these lines overlap and satisfy all conditions.
The objective function is also plotted as a line, and we adjust its position to identify the point in the feasible region that maximizes (or minimizes) its value.
  • First, plot the constraints on a graph.
  • Identify the feasible region where all constraints overlap.
  • Move the objective function line parallel to itself towards the optimal point in the feasible region.
This visual method makes it easier to deduce where the maximum revenue occurs and helps verify the solution derived through calculations.

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Most popular questions from this chapter

Defense Department Outlays The table shows the total national outlays \(y\) for defense functions \((\) in billions of dollars) for the years 2004 through 2011 . (Source: \(U . S .\) Office of Management and Budget) (a) Find the least squares regression line \(y=a t+b\) for the data, where \(t\) represents the year with \(t=4\) corresponding to \(2004,\) by solving the system for \(a\) and \(b .\) $$ \left\\{\begin{aligned} 8 b+60 a &=4700.5 \\ 60 b+492 a &=36,865.0 \end{aligned}\right. $$ (b) Use the regression feature of a graphing utility to confirm the result of part (a). (c) Use the linear model to create a table of estimated values of \(y .\) Compare the estimated values with the actual data. (d) Use the linear model to estimate the total national outlay for \(2012 .\) (e) Use the Internet, your school's library, or some other reference source to find the total national outlay for \(2012 .\) How does this value compare with your answer in part (d)? (f) Is the linear model valid for long-term predictions of total national outlays? Explain.

Graphical Reasoning Two concentric circles have radii \(x\) and \(y,\) where \(y>x .\) The area between the circles is at least 10 square units. (a) Find a system of inequalities describing the constraints on the circles. (b) Use a graphing utility to graph the system of inequalities in part (a). Graph the line \(y=x\) in the same viewing window. (c) Identify the graph of the line in relation to the boundary of the inequality. Explain its meaning in the context of the problem.

Ticket Sales For a concert event, there are \(\$ 30\) reserved seat tickets and \(\$ 20\) general admission tickets. There are 2000 reserved seats available, and fire regulations limit the number of paid ticket holders to \(3000 .\) The promoter must take in at least \(\$ 75,000\) in ticket sales. Find and graph a system of inequalities describing the different numbers of tickets that can be sold.

Choosing a Solution Method In Exercises \(49-56\) , solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{aligned} x-2 y &=1 \\ y &=\sqrt{x-1} \end{aligned}\right.$$

Solving a Linear Programming Problem, find the minimum and maximum values of the objective function and where they occur, subject to the indicated constraints. (For each exercise, the graph of the region determined by the constraints is provided.) $$ \begin{array}{c}{\text { Objective function: }} \\ {z=40 x+45 y} \\ {\text { Constraints: }} \\ {x \geq 0} \\ {y \geq 0}\end{array} $$ $$ \begin{array}{l}{8 x+9 y \leq 7200} \\ {8 x+9 y \geq 3600}\end{array} $$
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