application

application - Applications of Linear Systems Matrices...

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1 Systems of Linear Equations & Matrices Applications: Networks Balancing Equations Applications: Networks, Balancing Equations, Resource Allocation, Markov Chains Applications of Linear Systems & Matrices Applying Vectors & Matrices to help solve problems requires interpreting the problem in the notation of m t i e matrices Allocation of resources subject to a set of constraints Balancing chemical equations Network analysis (flow of fluid, traffic, data, goods & services through economic network etc.) Electrical networks (current, voltage – covered in physics) P d i Production costs Formulation of diets for humans, animals Simple Example: Shopping for fruit Simple Example: Shopping for fruit

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2 Allocation of Resources: Bacteria in Tube A biologist has placed three strains of bacteria (I,II,III) in a test tube, where they will feed on three different food ( C) h d 2300 f 800 f sources (A,B,C). Each day 2300 units of A, 800 units of B and 1500 units of C are placed in the test tube, and each bacterium consumes a certain number of units of each food per day, as shown. How many bacteria of each strain can coexist in the test tube and consume all of the food? Bacteria Bacteria Bacteria Strain I Strain II Strain III Units of Food A 2 2 4 Units of Food B 1 2 0 Units of Food C 1 3 1 Allocation of Resources: Bacteria in Tube, Again Same problem as before, but now the daily consumption of food (in units per day) is as consumption of food (in units per day) is as shown in the table. Assume 1500 units of A, 3000 units of B and 4500 units of C are placed in the test tube each day. Bacteria Strain I Bacteria Strain II Bacteria Strain III Units of Food A 1 1 1 Units of Food B 1 2 3 Units of Food C 1 3 5
3 Cambridge University Diet simplified version of a diet to achieve the desired amounts and proportions of nutrients, with 3 Ingredients and Amount (g) of nutrients supplied per 100g of Daily Amount foodstuffs if possible, find a combination of ingredients to provide the exact amounts of nutrients required by the diet nutrients supplied per 100g of ingredient Required by Diet Nutrient Nonfat Milk Soy Flour Whey Protein 36 51 13 33 Carbs 52 34 74 45 Fat 0 7 1.1 3 Cambridge University Diet Balancing Chemical Equations (compare molecules, then numbers of each atom) The combustion of ammonia (NH 3 ) in oxygen produces nitrogen and water. Balancing Chemical Equations (compare molecules, then numbers of each atom)

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4 Ex: Burning Octane in oxygen (C 8 H 18 + O 2 CO 2 + H 2 O) Ex: Burning Octane in oxygen (C 8 H 18 + O 2 CO 2 + H 2 O) Network Analysis: conservation of flow Network consists of branches with nodes or junctions , labelled with the flow (the amount of , labelled with the (the amount of some commodity that flows along that branch in a certain direction) Ex: Describe the possible flows through the network of water pipes shown, where flow is in litres per minute.
5 Ex: Describe the possible flows through the network of water pipes shown, where flow is in litres per minute.

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