Linear Programming

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Linear Programming — Free MYP5 Mathematics (Extended) Practice Questions

1QuestionGraphing constraintsConcept Practice

5 marks~8 minCriterion A

The diagram shows a coordinate grid with the lines

y = 2x + 1

and

x + y = 6

drawn.

Write down the coordinates of the point where

y = 2x + 1

crosses the

y

-axis. [1]

Find the coordinates of the point of intersection of

y = 2x + 1

and

x + y = 6

. [2]

Region

R

satisfies all three inequalities simultaneously:

y \leq 2x + 1, \qquad x + y \leq 6, \qquad x \geq 0.

Justify whether the point

(4, 1)

lies in region

R

. [2]

Solutions

2QuestionCorner pointsAssessment Practice

6 marks~9 minCriterion B

The diagram shows the feasible regions for three systems of inequalities.

System A:

x \ge 0

y \ge 0

x + y \le 6

System B:

x \ge 0

y \ge 0

x + y \le 6

x \le 4

System C:

x \ge 0

y \ge 0

x + y \le 6

x \le 4

y \le 3

Write down the coordinates of every corner point of the feasible region for each system. [2]

Describe the pattern connecting the number of constraints to the number of corner points across Systems A, B, and C. [1]

A student adds the constraint

y \le 1

to System C to create System D. Justify whether the pattern identified in part (b) still holds for System D, identifying any new corner point(s) and explaining why the corner-point count changes by the amount it does. [3]

Solutions

3QuestionCorner pointsAssessment Practice

8 marks~12 minCriterion C

The feasible region of a linear programming problem is bounded by the constraints:

y \geq 0, \quad x \geq 0, \quad y \leq -x + 8, \quad y \leq -\tfrac{1}{2}x + 5

The diagram shows the feasible region with three of its corner points labelled:

A(0, 0)

B(0, 5)

, and

D(8, 0)

Find the coordinates of the fourth corner point

C

, where the lines

y = -x + 8

and

y = -\tfrac{1}{2}x + 5

intersect. [2]

An objective function is

P = 3x + 4y

. Evaluate

P

at each of the four corner points and hence state the coordinates of the corner point that gives the maximum value of

P

, giving the maximum value. [3]

A student claims: "The maximum of a linear objective function over a polygonal feasible region always occurs at a corner point, because the level lines

P = k

are parallel to each other and the last one to touch the region must do so at a vertex." Justify whether this reasoning is mathematically correct, referring to the gradient of the objective function's level lines and the geometry of the feasible region. [3]

Solutions

4QuestionInequalitiesAssessment Practice

10 marks~15 minCriterion D

A small business produces wooden chairs and tables. Each chair requires 2 hours of labour and 1 unit of wood. Each table requires 3 hours of labour and 2 units of wood. The business has at most 240 hours of labour and 160 units of wood available per week. The profit per chair is 40 dollars and per table is 60 dollars.

Define your variables and write down all constraints as inequalities. [2]

Construct the feasible region on the axes provided, labelling all boundary lines and vertices. [3]

Determine the production plan that maximises weekly profit, stating the maximum profit. [2]

Advise the business owner whether this linear programming model should be used to make weekly production decisions, identifying one limitation that affects its reliability. [3]

Solutions

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