Semester I 2012/2013 Kohort 1

UNIVERSITI TUN HUSSEIN ONN MALAYSIABWM 21403 – MATEMATIK IV

GROUP ASSIGNMENT NO.2

Q1 Find the root of that lies in the interval [0.7, 0.9] by using

Bisection method. Iterate until .

Q2 Organic Wave is a huge manufacturer of modern cosmetics for rejuvenate facial cream. In order to fulfill the industry quality requirement, the company takes three different ingredients A, B and C, to produce three different substance solutions - Sodium Hydroxide, Dimethicone Crosspolymer and Sodium Dehydroacetate. Ingredient A, B and C have to be dissolved in water separately before they react to form the cosmetics ingredients. Table Q2 below represents the amount of solution A, B and C needed with all units measured are in cubic centimeters:

Table Q2

Volume ofSubstance ingredientsolution

Ingredient A Ingredient B Ingredient CTotal weight of

substance solution

Sodium Hydroxide 1.5 3.6 5.3 25.07

Dimethicone Crosspolymer 2.5 4.3 2.4 22.36

Sodium Dehydroacetate 2.7 5.5 3.2 28.14

By assuming volume (in cubic centimeters) of the solution A, volume

(in cubic centimeters) of the solution B and volume (in cubic centimeters) of the solution C, the above problem can be formulated as

.

Find the volumes (in cubic centimeters) of the solutions containing A, B, and C that being used in above problem, by using Doolittle factorization method.

This assignment is to be submitted on the fifth meeting (15 December 2012)

Semester I 2012/2013 Kohort 1

Answer: (TOTAL MARKS = 79)

Q1 M1 – apply Bisection method

M6 – calculate for each , , , , and

A36 – calculate for each cell of , , , , and

0 0.7 0.8 0.9 0.192 -0.019 -0.235

1 0.700 0.750 0.800 0.192 0.086 -0.019

2 0.750 0.775 0.800 0.086 0.034 -0.019

3 0.775 0.788 0.800 0.034 0.006 -0.019

4 0.788 0.794 0.800 0.006 -0.007 -0.019

5 0.788 0.791 0.794 0.006 0 -0.007

Since , therefore the root of is 0.791 M1 – stopping criteria, A1 – final answer

(45 marks)

Q2 The system of linear equations can be written in matrix form:

Step 1: From , solve for and .

A2 – for each L and U M1 – matrix multiplication

Semester I 2012/2013 Kohort 1

Compare element by element: M1 – comparing values

A9 – correct answers for each a, b, c, d, e, f, g, h and i

M6 – method of finding a, b, c, g, h and I only

.

M2 – writing down in terms of A=LU

Step 2 : From , solve for by forward substitution.

.

M2 – substitution in LY=b A3 – correct answers for each y

Step 3 : From , solve for by backward substitution.

.

M2 – substitution in Ux=Y A3 – correct answers for each x

Therefore, Organic Wave has to use 1.498 cm3 of ingredient A, 3.101 cm3 of ingredient B and 2.2 cm3 of ingredient C in order to form the substance solution as the cosmetics ingredient of rejuvenate facial cream.

M3 – giving answers in the point of application view

(34 marks)