Linear Programming

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Linear Programming — Free MYP4 Mathematics (Extended) Practice Questions

1QuestionCorner pointsConcept Practice

5 marks~8 minCriterion A

The diagram shows the feasible region defined by the constraints

x \ge 0

y \ge 0

x + y \le 6

, and

2x + y \le 10

, with corner points labelled

A

B

C

, and

D

Find the coordinates of corner point

C

, the intersection of

x + y = 6

and

2x + y = 10

, showing a complete algebraic method. [2]

An objective function is defined as

P = 3x + 5y

. Calculate the value of

P

at each of the four corner points

A(0,0)

B(5,0)

C

, and

D(0,6)

, and hence justify which corner point maximises

P

, stating the maximum value. [3]

Solutions

2QuestionInequalitiesAssessment Practice

6 marks~9 minCriterion B

The diagram shows the feasible region

R

defined by the three inequalities

x \geq 2, \quad y \geq 1, \quad x + y \leq 10,

where

x

and

y

are positive integers.

Write down the number of integer-coordinate points that lie on the boundary line

x + y = 10

within region

R

. [1]

For each integer value of

x

from

x = 2

x = 9

, find the total number of integer-coordinate points in region

R

. [2]

The column counts for

x = 2, 3, 4, \ldots, 9

form a sequence. Describe the pattern in this sequence and justify whether the general rule

C(n) = 10 - n

, for

2 \leq n \leq 9

, correctly models the number of valid integer points in column

x = n

. [3]

Solutions

3QuestionObjective functionAssessment Practice

6 marks~9 minCriterion C

A factory produces two products, X and Y. Each unit of X earns a profit of 3 dollars and each unit of Y earns a profit of 2 dollars. Daily production is subject to the following constraints, where

x

is the number of units of X and

y

is the number of units of Y:

x + y \leq 8

2x + y \leq 12

x \geq 0, \quad y \geq 0

The profit function is

P = 3x + 2y

(in dollars).

Write down the coordinates of the four vertices of the feasible region. [2]

Find the maximum daily profit, stating the values of

x

and

y

that achieve it. [2]

The factory manager proposes producing 5 units of X and 3 units of Y. Advise the manager whether this production plan should be adopted, justifying your answer with reference to the constraints and the optimal solution. [2]

Solutions

4QuestionInequalitiesAssessment Practice

10 marks~15 minCriterion D

A small business produces two types of wooden toys: chairs (

x

units per week) and tables (

y

units per week). Each chair requires 2 board-feet of wood and 3 hours of labour; each table requires 4 board-feet of wood and 2 hours of labour. Weekly resources are limited to at most 120 board-feet of wood and 90 hours of labour. A contract requires at least 10 chairs per week. Profit is 40 dollars per chair and 60 dollars per table.

State the system of inequalities that models this situation. [2]

Construct the feasible region and state the coordinates of all corner points. [3]

Deduce the production mix that maximises weekly profit, showing your reasoning. [2]

Justify whether the optimal solution of 15 chairs and 22.5 tables per week is operationally viable for the business, referring to two assumptions of the linear programming model. [3]

Solutions

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