Problem 22 - College Algebra

Question

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

The student will find the number of computation problems and word problems that should be solved to maximize the score by solving the linear programming problem. The maximum score will be found by substituting these values into the objective function.

Step 1: Define Variables

Let \'c\' represent the number of computation problems and \'w\' represent the number of word problems.

Step 2: Formulate Constraints

Two constraints can be derived from the problem: 1) The sum of the time taken by both types of problems cannot exceed 40 minutes. In other words, 2c + 4w ≤ 40. 2) The total number of problems attempted cannot exceed 12. So, c + w ≤ 12.

Step 3: Construct Objective Function

The objective is to maximize the total score obtained from both types of problems. So, the objective function is Score = 6c + 10w.

Step 4: Setup Linear Programming Problem

The problem now is to maximize Score = 6c + 10w, subject to the constraints 2c + 4w ≤ 40 and c + w ≤ 12.

Step 5: Solve the Linear Programming Problem

Visualize the constraints on a graph and identify the feasible region. The maximum value of the objective function should be found at one of the vertices of this region. Simplex method or Graphical method can be used to identify the optimal solution.

Step 6: Interpret the Solution

From the solution obtained, the numbers 'c' and 'w' correspond to the number of computation and word problems respectively, that need to be solved in order to maximize the total score.