You are about to take a test that contains computation...

You are about to take a test that contains computation problems worth 6 points cach and word problems worth 10 points cach. 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?

You are about to take a test that contains computation problems worth 6 points cach and word problems worth 10 points cach. 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?

Key Concepts

Objective Function

This is the function that is to be maximized or minimized. In optimization problems, it quantifies the goal, such as maximizing a score or minimizing cost, by forming a linear combination of decision variables.

Linear Programming

A mathematical technique used to determine the best outcome in a model whose requirements are represented by linear relationships. It involves defining decision variables, an objective function, and constraints to find the optimal solution under given conditions.

Constraints

These are the limitations or restrictions placed on the decision variables, often expressed as linear inequalities. They represent real-world restrictions like time limits or resource availability and define the boundaries of the possible solutions.

Feasible Region

The set of all possible points that satisfy all the constraints in a linear programming problem. The solution to the optimization problem is found within this region, and it is typically bounded by constraints.

Corner Point Principle

A key concept in linear programming, this principle states that if an optimal solution exists, it will occur at one of the vertices or 'corner points' of the feasible region. Evaluating the objective function at these points can lead to finding the optimum solution.

Integer Programming

A type of linear programming where some or all of the decision variables are restricted to integer values. This is crucial when the variables represent countable items, ensuring practical and applicable solutions in real-world problems.

Transcript

00:01 We are asked to solve a problem using linear optimization.

00:07 We're told that we're taking a test with computation problems that are six points each and word problems that are 10 points each.

00:16 We're told that we can complete a computation problem in two minutes and a word problem in four minutes.

00:25 And we're told that we have 40 minutes to take the test and can answer no more than 12 problems.

00:35 And assuming all our attempted.

00:40 Answers are going to be correct, we're asked to find how many of each type of problem that we should answer so that we maximize our score and to find the maximum score.

01:00 So let's define a few variables.

01:07 Let x be the number of computation problems attempted, and let y be the number of word problems attempted, and let z be the score, total score.

01:46 Well, we have that the total score is equal to the score from the attempted computation problems, which is going to be 6 times x plus the score from the word problems, which is 10 points per problem, 10 times y.

02:14 This is a linear function of x and y, and this is our objective function, which we are trying to maximize.

02:24 This function has some constraints on it.

02:31 So the time it takes to complete these problems is going to be the time to complete a computation problem or the computation problems, which is two minutes per problem.

02:52 So two times x plus the time required to complete the word problems, which is 4 times y and we're told that this total time must be less than or equal to 40 minutes.

03:28 The number of problems answered is simply number of computation problems x answered plus the number of word problems answered y and we're told that this must be less than or equal to 12 answers.

03:52 And so we have these constraints on our objective function and one other the other constraint, which is implicit in the problem statement, is that x and y are going to be greater than or equal to 0.

04:11 This is because we can't answer a negative number of questions.

04:16 We can't be adding questions to the test.

04:21 And so because x is greater than equal to y, and x and y are greater than or equal to zero follows that the solution to this system lies in the first quadrant.

04:39 I'll graph the first inequality in red.

04:43 To do this, i'll graph 2x plus 4y equals 40.

04:48 If y is equal to 0, we have that x is equal to 20.

04:51 So we have an x intercept at 20, 0...

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