Problem 22 - Algebra and Trigonometry

Question

Use the two steps for solving a linear programming problem, given in the box on page \(888,\) to solve the problems. You are about to take a test that contains computation problems worth 6 points each and word problems worth 10 points each. You can do a computation problem in 2 minutes and a word problem in 4 minutes. You have 40 minutes to take the test and may answer no more than 12 problems. Assuming you answer all the problems attempted correctly, how many of each type of problem must you answer to maximize your score? What is the maximum score?

Step-by-Step Solution

Answer

To maximize the score, the optimal strategy is to answer 4 computation problems and 8 word problems. The maximum achievable score is 100 points.

Step 1: Define the Variables

Let's denote 'C' as the number of computation problems and 'W' as the number of word problems that one answers.

Step 2: Formulate the Objective Function

The aim is to maximize the test score. And the score for each computation problem is 6 points, and for each word problem is 10 points. Thus, the objective function, denoted as 'Z', that needs to be maximized is: Z = 6C + 10W.

Step 3: Formulate the Constraints

The constraints are established by the conditions in the problem. One is that the total time spent on problems cannot exceed 40 minutes (where computation problems takes 2 minutes to solve and word problems take 4 minutes). This can be expressed as: 2C + 4W ≤ 40. The other constraint is that one can answer no more than 12 problems, presented as: C + W ≤ 12.

Step 4: Solve Linear Programming Problem

The problem is now a linear programming issue, which can be solved using different methods like the graphical method, Simplex method etc. The corner points of the feasible region are calculated and plugged into the objective function to find out which combination of computation and word problems maximize the score. Let's keep it simple and use the method of substitution. By substituting different feasible (integer) values of C and W in the objective function, it's found that maximum Z = 100 is achieved at C = 4 and W = 8.

Key Concepts

Objective Function

linear programming often involves an objective function, which is a mathematical expression that we aim to maximize or minimize. In the context of the exercise, the objective function represents the total score you can get on the test. We define it by considering the points awarded for each type of problem. Computation problems give 6 points each, while word problems provide 10 points each, resulting in the objective function:

• \( Z = 6C + 10W \)

Here, \( Z \) is the score, \( C \) stands for the number of computation problems solved, and \( W \) stands for the number of word problems solved. By finding the maximum value of \( Z \), you determine the best score possible within the given constraints.

Constraints

Constraints are essential in linear programming as they define the limitations or restrictions on the decision variables. In our test problem, there are two main constraints to observe.
The first constraint limits the time you can spend solving the problems. You cannot exceed 40 minutes in total.

• It translates mathematically to \( 2C + 4W \leq 40 \).

The second constraint deals with the maximum number of problems you can solve, which is 12 in this case.

• This is represented as \( C + W \leq 12 \).

Understanding these constraints helps in determining the feasible region. They are the rules you must follow while trying to optimize the objective function.

Feasible Region

The feasible region in linear programming is a graphical representation of all possible solutions that satisfy the constraints. It usually appears as a polygon or shape on a graph. In this exercise, the feasible region will show every possible combination of computation and word problems that fit within our conditions.
The constraints we have set: \( 2C + 4W \leq 40 \) and \( C + W \leq 12 \), along with the non-negativity conditions \( C \geq 0 \) and \( W \geq 0 \), bound this region.
All solutions within this region are viable. By plotting these constraints on a graph and shading the region of feasible solutions, you can visually identify where \( C \) and \( W \) overlap and explore combinations that meet the constraints.

Optimization

Optimization in linear programming involves finding the best outcome, typically the maximum or minimum value of the objective function, within the feasible region. In this scenario, you want to achieve the maximum test score by solving the best combination of computation and word problems within the given constraints.
To locate the optimal solution, test the objective function at each vertex or corner point of the feasible region. Each point represents a potential solution, showing the maximum score if certain numbers of computation and word problems are solved.
Finally, by observing the optimal value, which occurs at the vertex (4, 8), you find the strategic balance—solving 4 computation problems and 8 word problems yields a maximum score of 100 points. This process illustrates how linear programming effectively helps in making the best decisions based on specific criteria and limits.