0.10 Linear programming  (Page 4/4)

If we now substitute these points into the objective function, we get the following:

Therefore Mrs Nkosi makes the most profit if she plants $66\frac{2}{3}$ m ${}^2$ of mielies and $33\frac{1}{3}$ m ${}^2$ of potatoes. Her profit is R89 997.

End of chapter exercises

1. You are given a test consisting of two sections. The first section is on Algebra and the second section is on Geometry. You are not allowed to answer more than 10 questions from any section, but you have to answer at least 4 Algebra questions. The time allowed is not more than 30 minutes. An Algebra problem will take 2 minutes and a Geometry problem will take 3 minutes each to solve. If you answer $x_A$ Algebra questions and $y_G$ Geometry questions,
   1. Formulate the constraints which satisfy the above constraints.
   2. The Algebra questions carry 5 marks each and the Geometry questions carry 10 marks each. If $T$ is the total marks, write down an expression for $T$ .
2. A local clinic wants to produce a guide to healthy living. The clinic intends to produce the guide in two formats: a short video and a printed book. The clinic needs to decide how many of each format to produce for sale. Estimates show that no more than 10 000 copies of both items together will be sold. At least 4 000 copies of the video and at least 2 000 copies of the book could be sold, although sales of the book are not expected to exceed 4 000 copies. Let $x_v$ be the number of videos sold, and $y_b$ the number of printed books sold.
   1. Write down the constraint inequalities that can be deduced from the given information.
   2. Represent these inequalities graphically and indicate the feasible region clearly.
   3. The clinic is seeking to maximise the income, $I$ , earned from the sales of the two products. Each video will sell for R50 and each book for R30. Write down the objective function for the income.
   4. What maximum income will be generated by the two guides?
3. A patient in a hospital needs at least 18 grams of protein, 0,006 grams of vitamin C and 0,005 grams of iron per meal, which consists of two types of food, A and B. Type A contains 9 grams of protein, 0,002 grams of vitamin C and no iron per serving. Type B contains 3 grams of protein, 0,002 grams of vitamin C and 0,005 grams of iron per serving. The energy value of A is 800 kilojoules and of B 400 kilojoules per serving. A patient is not allowed to have more than 4 servings of A and 5 servings of B. There are $x_A$ servings of A and $y_B$ servings of B on the patient's plate.
   1. Write down in terms of $x_A$ and $y_B$
      1. The mathematical constraints which must be satisfied.
      2. The kilojoule intake per meal.
   2. Represent the constraints graphically on graph paper. Use the scale 1 unit = 20mm on both axes. Shade the feasible region.
   3. Deduce from the graphs, the values of $x_A$ and $y_B$ which will give the minimum kilojoule intake per meal for the patient.
4. A certain motorcycle manufacturer produces two basic models, the 'Super X' and the 'Super Y'. These motorcycles are sold to dealers at a profit of R20 000 per 'Super X' and R10 000 per 'Super Y'. A 'Super X' requires 150 hours for assembly, 50 hours for painting and finishing and 10 hours for checking and testing. The 'Super Y' requires 60 hours for assembly, 40 hours for painting and finishing and 20 hours for checking and testing. The total number of hours available per month is: 30 000 in the assembly department, 13 000 in the painting and finishing department and 5 000 in the checking and testing department. The above information can be summarised by the following table:

   Department	Hours for `Super X'	Hours for Super `Y'	Maximum hours available per month
   Assembley	150	60	30 000
   Painting and Finishing	50	40	13 000
   Checking and Testing	10	20	5 000

   Let $x$ be the number of 'Super X' and $y$ be the number of 'Super Y' models manufactured per month.
   1. Write down the set of constraint inequalities.
   2. Use the graph paper provided to represent the constraint inequalities.
   3. Shade the feasible region on the graph paper.
   4. Write down the profit generated in terms of $x$ and $y$ .
   5. How many motorcycles of each model must be produced in order to maximise the monthly profit?
   6. What is the maximum monthly profit?
5. A group of students plan to sell $x$ hamburgers and $y$ chicken burgers at a rugby match. They have meat for at most 300 hamburgers and at most 400 chicken burgers. Each burger of both types is sold in a packet. There are 500 packets available. The demand is likely to be such that the number of chicken burgers sold is at least half the number of hamburgers sold.
   1. Write the constraint inequalities.
   2. Two constraint inequalities are shown on the graph paper provided. Represent the remaining constraint inequalities on the graph paper.
   3. Shade the feasible region on the graph paper.
   4. A profit of R3 is made on each hamburger sold and R2 on each chicken burger sold. Write the equation which represents the total profit, P, in terms of $x$ and $y$ .
   5. The objective is to maximise profit. How many, of each type of burger, should be sold to maximise profit?
6. Fashion-cards is a small company that makes two types of cards, type X and type Y. With the available labour and material, the company can make not more than 150 cards of type X and not more than 120 cards of type Y per week. Altogether they cannot make more than 200 cards per week. There is an order for at least 40 type X cards and 10 type Y cards per week.Fashion-cards makes a profit of R5 for each type X card sold and R10 for each type Y card. Let the number of type X cards be $x$ and the number of type Y cards be $y$ , manufactured per week.
   1. One of the constraint inequalities which represents the restrictions above is $x\le 150$ . Write the other constraint inequalities.
   2. Represent the constraints graphically and shade the feasible region.
   3. Write the equation that represents the profit P (the objective function), in terms of $x$ and $y$ .
   4. Calculate the maximum weekly profit.
7. To meet the requirements of a specialised diet a meal is prepared by mixing two types of cereal, Vuka and Molo . The mixture must contain $x$ packets of Vuka cereal and $y$ packets of Molo cereal. The meal requires at least 15 g of protein and at least 72 g of carbohydrates. Each packet of Vuka cereal contains 4 g of protein and 16 g of carbohydrates. Each packet of Molo cereal contains 3 g of protein and 24 g of carbohydrates. There are at most 5 packets of cereal available. The feasible region is shaded on the attached graph paper.
   1. Write down the constraint inequalities.
   2. If Vuka cereal costs R6 per packet and Molo cereal also costs R6 per packet, use the graph to determine how many packets of each cereal must be used for the mixture to satisfy the above constraints in each of the following cases:
      1. The total cost is a minimum.
      2. The total cost is a maximum (give all possibilities).

   

8. A bicycle manufacturer makes two different models of bicycles, namely mountain bikes and speed bikes. The bicycle manufacturer works under the following constraints: No more than 5 mountain bicycles can be assembled daily.No more than 3 speed bicycles can be assembled daily. It takes one man to assemble a mountain bicycle, two men to assemble a speed bicycle and there are 8 men working at the bicycle manufacturer.Let $x$ represent the number of mountain bicycles and let $y$ represent the number of speed bicycles.
   1. Determine algebraically the constraints that apply to this problem.
   2. Represent the constraints graphically on the graph paper.
   3. By means of shading, clearly indicate the feasible region on the graph.
   4. The profit on a mountain bicycle is R200 and the profit on a speed bicycle is R600. Write down an expression to represent the profit on the bicycles.
   5. Determine the number of each model bicycle that would maximise the profit to the manufacturer.