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# lecture8hout - MATH 006 Calculus and Linear Algebra(Lecture...

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MATH 006 Calculus and Linear Algebra (Lecture 8) Albert Ku HKUST Mathematics Department Albert Ku (HKUST) MATH 006 1 / 22

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Outline 1 Applications of Gauss-Jordan Elimination 2 Tank Car Leases 3 Traffic Flow 4 Boat Production Albert Ku (HKUST) MATH 006 2 / 22
Applications of Gauss-Jordan Elimination Applications of Gauss-Jordan Elimination A lot of problems in real life can be modelled by systems of linear equations. In this lecture, we will study three different application problems which can be solved by applying Gauss-Jordan elimination on the systems of linear equations that formulate the problems. Albert Ku (HKUST) MATH 006 3 / 22

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Tank Car Leases Tank Car Leases Example A chemical manufacturer wants to lease a fleet of 24 railroad tank cars with a combined carrying capacity of 520,000 gallons. Tank cars with three different carrying capacities are available: 8,000 gallons, 16,000 gallons, and 24,000 gallons. How many of each type of tank car should be leased? Albert Ku (HKUST) MATH 006 4 / 22
Tank Car Leases Solution To formulate the problem, we first need to define relevant quantities as variables: Let x 1 be the number of tank cars with a capacity of 8,000 gallons Let x 2 be the number of tank cars with a capacity of 16,000 gallons Let x 3 be the number of tank cars with a capacity of 24,000 gallons Given the criteria stated in the problem, we can obtain the following linear system: ( x 1 + x 2 + x 3 = 24 8000 x 1 + 16000 x 2 + 24000 x 3 = 520000 The corresponding augmented matrix is 1 1 1 24 8000 16000 24000 520000 Albert Ku (HKUST) MATH 006 5 / 22

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Tank Car Leases 1 1 1 24 8000 16000 24000 520000 1 8000 R 2 R 2 -→ 1 1 1 24 1 2 3 65 R 2 +( - 1) R 1 R 2 -→ 1 1 1 24 0 1 2 41 R 1 +( - 1) R 2 R 1 -→ 1 0 - 1 - 17 0 1 2 41 The linear system corresponding to the reduced form is x 1 - x 3 = - 17 x 2 + 2 x 3 = 41 Albert Ku (HKUST) MATH 006 6 / 22
Tank Car Leases Since x 3 is the free variable, we let x 3 = t , where t is any real number. Then the general solution can be written as follows: x 1 = - 17 + t x 2 = 41 - 2 t x 3 = t where t is any real number.

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lecture8hout - MATH 006 Calculus and Linear Algebra(Lecture...

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