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Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203

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BuyFindarrow_forward

Finite Mathematics and Applied Cal...

7th Edition
Stefan Waner + 1 other
ISBN: 9781337274203
Textbook Problem

Transportation Scheduling Duffin House is about to start a promotional blitz for its new book, Physics for the Liberal Arts. The company has 20 salespeople stationed in Chicago and 10 in Denver, and it would like to fly 15 of them to sales fairs at each of Los Angeles and New York. A round-trip plane flight from Chicago to LA costs $200; from Chicago to NY costs $150; from Denver to LA costs $400; and from Denver to NY costs $200. For tax reasons, Duffin House needs to budget exactly $6,500 for the total cost of the plane flights. How many salespeople should the company fly from each of Chicago and Denver to each of LA and NY?

To determine

To calculate: The amount of salespeople that should fly from Denver and Chicago to each of Los Angeles and New York if a round-trip plane flight from Chicago to NY costs $150, Chicago to LA costs $200, Denver to NY costs $200 and Denver to LA costs $400.

Explanation

Given Information:

The promotional blitz for the book Physics for the Liberal Arts, is to be started by the Duffin House. There are 20 salespeople in Chicago and 10 in Denver working for the company. The company wants to fly 15 of the salespeople to the sales fair at each of the Los Angeles and New York. A round-trip plane flight from Chicago to NY costs $150, Chicago to LA costs $200, Denver to NY costs $200 and Denver to LA costs $400. Duffin House has to budget for $6500 for plane flights due to tax reasons.

Formula Used:

Elementary row operations:

Type 1: Replacing the row Ri by aRi, where a is a nonzero number.

Type 2: Replacing the row Ri by aRi±bRj, where a is a nonzero number.

Gauss Jordan reduction method:

Step 1: First clear the fractions or decimals if any.

Step 2: Pivot the first nonzero element of the first row.

Step 3: Use the pivot to clear its column.

Step 4: Pivot the first nonzero element in the second row and clear the column.

Step 5: Pivot the first nonzero element in the third row and clear the column.

Step 6: Turn all the pivot elements into a 1.

Calculation:

Firstly, identify and label the unknowns.

Considering the provided information, the unknowns should be labelled like this,

x represent the number of people flown from Chicago to LA

y represent the number of people flown from Chicago to NY

z represent the number of people flown from Denver to LA

w represent the number of people flown from and Denver to NY

It is known that the sales people available in Chicago and Denver respectively are 20 and 10. The conditions can be represented in the form of equations as,

Sales people available in Chicago: x+y=20

Sales people available in Denver: z+w=10

Also, the number of sales people required in both LA and NY is 15 each.

Sales people required in LA: x+z=15

Sales people required in NY: y+w=15

Now, we are provided that the total budget for plane flights is 6500, which can be represented in the form of equation as,

200x+150y+400z+200w=6500

The augmented matrix for the obtained equations is:

[1100200011101010150101152001504002006500]

Apply Gauss Jordan reduction method to get the solution of the given system of equation.

Begin by clearing the fraction in the rows of the matrix by the operation R4150R4

[1100200011101010150101152001504002006500][1100200011101010150101154384130]

Now, pivot the first nonzero element of first row and clear its column

Perform the operation R3R3R1,R5R54R1,

[1100200011101010150101154384130][11002000111001105010115018450]

Now, pivot the first nonzero element of second row and clear its column

Perform the operation R3R3R2,R5R58R2,

[11002000111001105010115018450][110020001110010115010115010430]

Now, select the first nonzero element of third row and clear its column.

Perform the operation R1R1+R3,R4R4+R3

[110020001110010115010115010430][1001500111001011500000010430]

Further apply the operation R5R5R3

[1001500111001011500000010430][1001500111001011500000000315]

Now, select the first nonzero element of fifth row and clear its column

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