The Collin Freight Company has an order for three products to be delivered to a
ID: 3037824 • Letter: T
Question
The Collin Freight Company has an order for three products to be delivered to a destination. Product I requires 10 cubic feet, weighs 10 pounds, and has a value of $100. Product II requires 8 cubic feet, weighs 20 pounds, and has a value of $20. Product III requires 20 cubic feet, weighs 40 pounds, and has a value of $200. If the carrier can carry 6,000 cubic feet, 11,000 pounds, and is insured for $36.900, how many of each product can be carried? a. Identify the variables of the problem using x, y and z. b. Fill out the following table from the information given: c. Write the augmented matrix for the system of equations. d. Solve the system of equations using Gauss-Jordan Elimination. (Attach work) e. In at least one complete sentence, answer the question: How many of each product can be carried?Explanation / Answer
(a). The variables of the problem are the number of the products I,II and III. Let these be x , y and z respectively.
(b)
x
y
z
Total
Volume (cu.ft.)
10
8
20
38
Weight (lb.)
10
20
40
70
Value ($)
100
20
200
320
(c) The 3 equations which can be arrived at from the conditions about the maximum weight and volume which can be carried by the carrier and the limit on insurance are as under:
10x+8y+20z 6000. On dividing both the sides by 2, we get 5x+4y+10z 3000…(1).
10x+20y+40z 11000. On dividing both the sides by 10, we get x +2y+4z 1100…(2).
100x +20y+200z 36900. On dividing both the sides by 20, we get 5x+y+10z 1845…(3)
The augmented matrix of the system is A =
5
4
10
3000
1
2
4
1100
5
1
10
1845
( we can create the augmented matrix of the system from the original inequalities also, but smaller numbers are easier to deal with).
(d) We can reduce A to its RREF, using Gauss-Jordan elimination as under:
Multiply the 1st row by 1/5
Add -1 times the 1st row to the 2nd row
Add -5 times the 1st row to the 3rd row
Multiply the 2nd row by 5/6
Add 3 times the 2nd row to the 3rd row
Multiply the 3rd row by 1/5
Add -5/3 times the 3rd row to the 2nd row
Add -2 times the 3rd row to the 1st row
Add -4/5 times the 2nd row to the 1st row
Then the RREF of A is
1
0
0
254
0
1
0
385
0
0
1
19
(e) Maximum 254 numbers of Product I, 385 numbers of Product II and 19 numbers of Product III can be carried.
x
y
z
Total
Volume (cu.ft.)
10
8
20
38
Weight (lb.)
10
20
40
70
Value ($)
100
20
200
320
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