Linear Algebra Winter 2018, MATH 261: Worksheet 1 Name 1. A chemical manufacture

Question

Linear Algebra

Winter 2018, MATH 261: Worksheet 1 Name 1. 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 capacities are available: 8,000 gallons, 16,000 gallons, and 24,000 gallons. How many of each type of tank car should be leased? (a) Define the variables 1, 2, and t3 using complete sentences. (b) Set up a system of equations based on the manufacturer's needs. There should be two equations. (c) Create an augmented matrix for the system of equations in part (b). (d) Row reduce the matrix from (c) into echelon form and then into re- duced echelon form.

Explanation / Answer

I am solving the first four parts as per Chegg guidelines, post multiple question to get the remaining parts

Q1)

a) Let the variables be

x1 = number of 8000 gallons tank car

x2 = number of 16000 gallons tank car

x2 = number of 24000 gallons tank car

b) Now the equations from the constraints will be

Constraint 1: Total number of tank cars is equal to 24

x1 + x2 + x3 = 24

Constraint 2: Combined carrying capacity constraint

8000x1 + 16000x2 + 24000x3 = 520000

Modifying the second equation by dividing it by 8000

x1 + 2x2 + 3x3 = 65

c) The augmented matrix will be

d)

Now reducing the matrix into row form and row-reduced echleon form

R2 -> R2 - R1

R1->R1-R2

Hence the equation will be

x1 - x3 = -17

x2 + 2x3 = 41

Now let us assume x3=t

x1 = -17 + t

x2 = 41 - 2t

Since x1,x2 and x3 can never be negative, hence the minimum value of t must be 17 and max value will be 20, otherwise x2 will become negative

1 1 1 24 1 2 3 65

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