You are not allowed to answer more than \(\text{10}\) questions from any section:

\begin{align*} x & \leq 10 \\ y & \leq 10 \end{align*}

You have to answer at least \(\text{4}\) algebra questions:

\begin{align*} x & \geq 4 \end{align*}

The time allowed is not more than \(\text{30}\) minutes. An algebra problem will take \(\text{2}\) minutes and a geometry problem will take \(\text{3}\) minutes to solve:

\begin{align*} 2x + 3y & \leq 30 \end{align*}

\begin{align*} T = & (\text{5 marks}) \times (\text{algebra questions answered}) + \\ & (\text{10 marks}) \times (\text{geometry questions answered}) \\ T = & 5x + 10y \end{align*}

Estimates show that no more than \(\text{10 000}\) copies of both items together will be sold:

\begin{align*} x + y & \leq \text{10 000} \end{align*}

At least \(\text{4 000}\) copies of the video could be sold:

\begin{align*} x & \geq \text{4 000} \end{align*}

At least \(\text{2 000}\) copies of the book could be sold:

\begin{align*} y & \geq \text{2 000} \end{align*}

Sales of the book are not expected to exceed \(\text{4 000}\) copies:

\begin{align*} y & \leq \text{4 000} \end{align*}

\begin{align*} I = & (\text{R}\,\text{50}) \times (\text{videos sold}) + \\ & (\text{R}\,\text{30}) \times (\text{printed books sold}) \\ I = & 50x + 30y \end{align*}

The vertices of the feasible region are as follows:

\begin{align*} \text{Point A: } & (\text{4 000};\text{2 000}) \\ \text{Point B: } & (\text{4 000};\text{4 000}) \\ \text{Point C: } & (\text{6 000};\text{4 000}) \\ \text{Point D: } & (\text{8 000};\text{2 000}) \end{align*}

The cost at each vertex is as follows:

\begin{align*} \text{Point A: } I & = 50(\text{4 000}) + 30(\text{2 000}) = \text{260 000} \\ \text{Point B: } I & = 50(\text{4 000}) + 30(\text{4 000}) = \text{320 000} \\ \text{Point C: } I & = 50(\text{6 000}) + 30(\text{4 000}) = \text{420 000} \\ \text{Point D: } I & = 50(\text{8 000}) + 30(\text{2 000}) = \text{460 000} \end{align*}

The maximum profit of \(\text{R}\,\text{460 000}\) can be made if \(\text{8 000}\) videos and \(\text{2 000}\) books are sold.

Adding the assembly hours:

\begin{align*} (150)x + (60)y & \geq \text{30 000} \end{align*}

Adding the painting and finishing:

\begin{align*} (50)x + (40)y& \geq \text{13 000} \end{align*}

Adding the checking and testing:

\begin{align*} (10)x + (20)y & \geq \text{5 000} \end{align*}

See above

\begin{align*} E = & (\text{profit per Super X}) \times (\text{number of Super X bikes sold}) + (\text{profit per Super Y}) \times (\text{number of Super Y bikes sold})\\ E = & (\text{20 000})x + (\text{10 000})y \end{align*}

The vertices of the feasible region are as follows:

\begin{align*} \text{Point A: } & (\text{0};\text{0}) \\ \text{Point B: } & (\text{200};\text{0}) \\ \text{Point C: } & (\text{140};\text{150}) \\ \text{Point D: } & (\text{100};\text{200}) \\ \text{Point E: } & (\text{0};\text{250}) \end{align*}

The cost at each vertex is as follows:

\begin{align*} \text{Point A: } E & = \text{20 000}(\text{0}) + \text{10 000}(\text{0}) = \text{0} \\ \text{Point B: } E & = \text{20 000}(\text{200}) + \text{10 000}(\text{0}) = \text{4 000 000} \\ \text{Point C: } E & = \text{20 000}(\text{140}) + \text{10 000}(\text{150}) = \text{4 300 000} \\ \text{Point D: } E & = \text{20 000}(\text{100}) + \text{10 000}(\text{200}) = \text{4 000 000} \\ \text{Point E: } E & = \text{20 000}(\text{0}) + \text{10 000}(\text{250}) = \text{2 500 000} \end{align*}

To maximize the objective function select Point C.

The maximum profit can be made if \(\text{140}\) Super X motorbikes and \(\text{150}\) Super Y motorbikes are sold.

The maximum possible profit is \(\text{R}\,\text{4 300 000}\).

They have meat for at most 300 hamburgers:

\begin{align*} x \leq \text{300} \end{align*}

They have meat for at most 400 chicken burgers:

\begin{align*} y \leq \text{400} \end{align*}

The demand is likely to be such that the number of chicken burgers sold is at least half the number of hamburgers sold:

\begin{align*} y & \geq (0.5) x \end{align*}

Each burger of both types is sold in a packet. There are 500 packets available:

\begin{align*} x + y & \leq 500 \end{align*}

\begin{align*} P = & (\text{profit per hamburger}) \times (\text{number of hamburgers sold}) + (\text{profit per chicken burger}) \times (\text{number of chicken burgers sold})\\ P = & (\text{3})x + (\text{2})y \end{align*}

The vertices of the feasible region are as follows:

\begin{align*} \text{Point A: } & (\text{0};\text{0}) \\ \text{Point B: } & (\text{300};\text{150}) \\ \text{Point C: } & (\text{300};\text{200}) \\ \text{Point D: } & (\text{100};\text{400}) \\ \text{Point E: } & (\text{0};\text{400}) \end{align*}

The cost at each vertex is as follows:

\begin{align*} \text{Point A: } P & = \text{3}(\text{0}) + \text{2}(\text{0}) = \text{0} \\ \text{Point B: } P & = \text{3}(\text{300}) + \text{2}(\text{150}) = \text{1 200} \\ \text{Point C: } P & = \text{3}(\text{300}) + \text{2}(\text{200}) = \text{1 300} \\ \text{Point D: } P & = \text{3}(\text{100}) + \text{2}(\text{400}) = \text{1 100} \\ \text{Point E: } P & = \text{3}(\text{0}) + \text{2}(\text{400}) = \text{800} \end{align*}

To maximize the objective function select Point C.

The maximum possible profit of \(\text{R}\,\text{1 300}\) can be made if \(\text{300}\) hamburgers and \(\text{200}\) chicken burgers are sold.

The company can not make more than \(\text{150}\) cards of type X per week: \[x \leq \text{150}\]

The company can make not more than \(\text{120}\) cards of type Y per week: \[y \leq \text{120}\]

Altogether they cannot make more than \(\text{200}\) cards per week: \[x + y \leq 200\]

There is an order for at least \(\text{40}\) type X cards and \(\text{10}\) type Y cards per week: \begin{align*} x & \geq 40 \\ y & \geq 10 \end{align*}

\begin{align*} P = & (\text{profit per card X}) \times (\text{number of card X sold}) + (\text{profit per card Y}) \times (\text{number of card Y sold})\\ P = & (\text{5})x + (\text{10})y \end{align*}

The vertices of the feasible region are as follows:

\begin{align*} \text{Point A: } & (\text{40};\text{10}) \\ \text{Point B: } & (\text{150};\text{10}) \\ \text{Point C: } & (\text{150};\text{50}) \\ \text{Point D: } & (\text{80};\text{120}) \\ \text{Point E: } & (\text{40};\text{120}) \end{align*}

The cost at each vertex is as follows:

\begin{align*} \text{Point A: } P & = \text{5}(\text{40}) + \text{10}(\text{10}) = \text{300} \\ \text{Point B: } P & = \text{5}(\text{150}) + \text{10}(\text{10}) = \text{850} \\ \text{Point C: } P & = \text{5}(\text{150}) + \text{10}(\text{50}) = \text{1 250} \\ \text{Point D: } P & = \text{5}(\text{80}) + \text{10}(\text{120}) = \text{1 600} \\ \text{Point E: } P & = \text{5}(\text{40}) + \text{10}(\text{120}) = \text{1 400} \end{align*}

To maximize the objective function select Point D.

The maximum possible weekly profit of \(\text{R}\,\text{1 600}\) can be made if \(\text{80}\) of card X and \(\text{120}\) of card Y are sold.

Adding the protein:

\begin{align*} (4)x + (3)y \geq \text{15} \end{align*}

Adding the carbohydrates:

\begin{align*} (16)x + (24)y \geq \text{72} \end{align*}

There are at most 5 packets of cereal available:

\begin{align*} x + y & \leq 5 \end{align*}

The objective function can be written as follows:

\begin{align*} P = & (\text{profit per packet of Vuka}) \times (\text{number of packets of Vuka}) + (\text{profit per packet of Molo}) \times (\text{number of packets of Molo})\\ P = & (\text{6})x + (\text{6})y \end{align*}

The vertices of the feasible region are as follows:

\begin{align*} \text{Point A: } & (\text{3};\text{1}) \\ \text{Point B: } & (\text{4,5};\text{0}) \\ \text{Point C: } & (\text{5};\text{0}) \\ \text{Point D: } & (\text{0};\text{5}) \end{align*}

The cost at each vertex is as follows:

\begin{align*} \text{Point A: } P & = \text{6}(\text{3}) + \text{6}(\text{1}) = \text{24} \\ \text{Point B: } P & = \text{6}(\text{4,5}) + \text{6}(\text{0}) = \text{27} \\ \text{Point C: } P & = \text{6}(\text{5}) + \text{6}(\text{0}) = \text{30} \\ \text{Point D: } P & = \text{6}(\text{0}) + \text{6}(\text{5}) = \text{30} \end{align*}

To minimize the objective function select Point A. The minimum possible cost of \(\text{R}\,\text{24}\) can be made if \(\text{3}\) packets of Vuka and \(\text{1}\) packets of Molo are sold.

To maximize the objective function select Point C, or Point D. The maximum possible cost of \(\text{R}\,\text{30}\) can be made if either \(\text{5}\) packets of Vuka and \(\text{0}\) packets of Molo are sold, or \(\text{0}\) packets of Vuka and \(\text{5}\) packets of Molo are sold.