New Zealand
Level 8 - NCEA Level 3

# Linear Programming - Applications

## Interactive practice questions

A manufacturer produces two types of tables. Each table requires a cabinet maker and a painter to build. The time taken for each worker varies according to the table below.

Let $x$x represent the number of round tables built in a week, and let $y$y represent the number of square tables built in a week.

 Cabinet Maker Painter Round Table $2$2 hours $2$2 hours Square Table $2$2 hours $1$1 hours Total time available in a week $34$34 hours $30$30 hours
a

Finish the set of constraint inequalities below:

1. $x\ge0$x0
2. $y\ge0$y0
3. $\editable{}+\editable{}\le34$+34
4. $\editable{}+\editable{}\le\editable{}$+
b

Graph the region defined by inequalities 3 and 4 in the first quadrant:

c

The region of possible solutions that you graphed in part (b) has been sketched more clearly below:

List the four vertices of the feasible region:

$\left(0,0\right),\left(\editable{},\editable{}\right),\left(\editable{},\editable{}\right),\left(\editable{},\editable{}\right)$(0,0),(,),(,),(,)

d

If a round table sells for a profit of $\$250$$250 and a square table makes a profit of \200$$200, write the objective function that models the weekly profit $z$z.

e

Complete the table below to determine the profit corresponding to each vertex of the feasible region:

 Vertex Profit (dollars) $\left(0,0\right)$(0,0) $0$0 $\left(0,17\right)$(0,17) $\editable{}$ $\left(13,4\right)$(13,4) $\editable{}$ $\left(15,0\right)$(15,0) $\editable{}$
f

Complete the following sentences:

The maximum weekly profit is $\editable{}$ dollars, and occurs at the point $\left(\editable{},\editable{}\right)$(,).

That is to say, $\editable{}$ square tables and $\editable{}$ round tables should be made each week to achieve the maximum profit.

Easy
Approx 9 minutes

A shoe manufacturer produces shoes for both men and women. Each pair of shoes is made of rubber for the soles and leather for the upper. The amount of material required to make each type is shown in the table below.

Let $x$x represent the number of pairs of women's shoes made in a week, and let $y$y represent the number of pairs of men's shoes made in a week.

The popular shop Bergner's Burgers makes and sells two types of burgers; beef and chicken. Each day, Mr Bergner knows that they need to make at least $190$190 beef burgers and $170$170 chicken burgers. The maximum number of burgers that can be made in one day is $460$460.

Let $x$x represent the number of beef burgers made per day, and let $y$y represent the number of chicken burgers made in a day.

A bakery produces two types of cake, each using the three main ingredients flour, sugar and butter in different proportions. The bakery has a certain amount of each type of ingredient delivered each week. The requirements for each cake and the amounts that can be stored are shown in the table below.

Let $x$x represent the number of type $A$A cakes baked in a week, and let $y$y represent the number of type $B$B cakes baked in a week.

### Outcomes

#### M8-4

Use curve fitting, log modelling, and linear programming techniques

#### 91574

Apply linear programming methods in solving problems