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University of the Witwatersrand, Johannesburg
MATH3033
Course or topic No(s)
Course or topic name(s)
Paper Number & title
Mathematics III (Engineering)
Examination/Test∗ to be
held during month(s) of
March 2016
(∗ delete as applicable)
Year of Study
(Art & Sciences leave blank)
Third
Degrees/Diplomas for which
this course is prescribed
(BSc (Eng) should indicate which branch)
B.Sc (Eng) (Industrial)
Faculty/ies presenting
candidates
Internal examiners
and telephone
number(s)
Dr S Jamal x76219
External examiner(s)
Calculator policy
Time allowance
Instructions to candidates
(Examiners may wish to use
this space to indicate, inter alia,
the contribution made by this
examination or test towards
the year mark, if appropriate)
Scientific calculators are allowed
Course
Nos
None
Hours
Answer ALL questions.
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Approximate marks are shown.
An information sheet is attached.
Internal Examiners or Heads of Department are requested to sign the declaration overleaf
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MATH3033 – Mathematics III (Engineering)
March 2016
1. As the Internal Examiner/Head of Department, I certify that this question paper is in final
form, as approved by the External Examiner, and is ready for reproduction.
2. As the Internal Examiner/Head of Department, I certify that this question paper is in final
form and is ready for reproduction.
(1. is applicable to formal examinations as approved by an external examiner, while 2. is
applicable to formal tests not requiring approval by an external examiner—Delete whichever is
not applicable)
Name:
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(Page 2 of 2)
MATH3033 – Mathematics III (Engineering)
March 2016
(Page 1 of 2)
Question 1
Suppose that two basic types of food are supplied to soldiers in an army: meats and potatoes.
Each kilogram of meats costs R1, while each kilogram of potatoes costs R0.25. To minimize
expenses, army officials consider serving only potatoes. However, there are some basic nutritional
requirements that call for meats in a soldiers diet. In particular, each soldier should get at least
400 grams of carbohydrates, 40 grams of dietary fiber, and 200 grams of protein in their daily
diet. Nutrients offered per kilogram of each of the two types of food, as well as the daily
requirements, are provided in the following table:
Nutrients
carbohydrates
dietary fibre
protein
Meats
40 grams
5 grams
100 grams
Potatoes
200 grams
40 grams
20 grams
Daily Requirement
400 grams
40 grams
200 grams
By graphing the feasible region (or otherwise), determine a minimal cost diet comprised of
meats and potatoes that satisfies the nutritional requirements. Let x1 the number of kilograms
of meat and let x2 be the the number of kilograms of potatoes. How much is this minimal cost
per soldier?
(12 marks)
Question 2
a. Prove Theorem 1 from the attached information sheet.
(4 marks)
b. Hence prove Corollary 2 from the attached information sheet.
(3 marks)
Question 3
Woody Inc. manufactures two types of wooden toys: soldiers and trains. Each soldier contributes
R3 to the profit while each train contributes R2. The manufacture of wooden soldiers and trains
requires two types of skilled labor: carpentry and finishing. A soldier requires 2 hours of finishing
labour and 1 hour of carpentry labour. A train requires 1 hour of finishing labour and 1 hour
of carpentry labour. Each week, only 100 finishing hours and 80 carpentry hours are available.
Demand for trains is unlimited, but at most 40 soldiers are bought each week. If x1 is the
number of soldiers produced and x2 the number of trains produced each week, then Woody
maximizes their weekly profit according to the LP:
Maximise 3x1 + 2x2 subject to x1 , x2 ≥ 0 and
2x1 + x2 ≤ 100
x1 + x2 ≤ 80
x1 ≤ 40.
(i) State the dual problem.
(ii) If (x1 , x2 ) = (20, 60) is the optimal solution to the primal problem, use the Equilibrium
Theorem (complementary slackness conditions) to solve the dual.
(iii) Interpret and explain the dual solution in terms of shadow prices for this problem.
(20 marks)
MATH3033 – Mathematics III (Engineering)
March 2016
(Page 2 of 2)
Question 4
Consider the tableau
x1 x2 x3
y1 −3
3
1
y2
2 −1 −2
y3 −1
0
1
which represents a system of equations that express x1 , x2 and x3 in terms of y1 , y2 and y3 .
Pivot to swap x3 and y3 ; write out the new tableau.
(6 marks)
(Grand Total: 45 marks)