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Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

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BuyFindarrow_forward

Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

APPLICATIONS

Nutrition A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home the patient finds three brands of vitamin pills. The amounts of the relevant vitamins per pill are given in the table. How many pills of each type should he take every day to get 50 mg of each vitamin?

VitaMax Vitron VitaPlus
Niacin (mg) 5 10 15
Riboflavin (mg) 15 20 0
Thiamin (mg) 10 10 10

To determine

To find:

The number of pills required to get 50 mg of each vitamin.

Explanation

Given:

A doctor recommends that a patient take 50 mg each of niacin, riboflavin, and thiamin daily to alleviate a vitamin deficiency. In his medicine chest at home, the patient finds three brands of vitamin pills. The following table gives the amounts of the relevant vitamins per pill:

VitaMax Vitron VitaPlus
Niacin (mg) 5 10 15
Riboflavin (mg) 15 20 0
Thiamin (mg) 10 10 10

Table (1)

Approach:

Let x, y, and z represents the number of pills in VitaMax, Vitron, and VitaPlus respectively, to get 50 mg of each vitamin.

Write the augmented matrix of the system.

Use elementary row operations to change the augmented matrix to row-echelon form.

Write the new system of equations that corresponds to the row-echelon form of the augmented matrix and solve by back-substitution.

Calculation:

Since the vitamins required is 50 mg.

The system of equations is given below.

5x+10y+15z=5015x+20y=5010x+10y+10z=50

Consider the system of equations {5x+10y+15z=5015x+20y=5010x+10y+10z=50.

The augmented matrix is given by,

[5101550152005010101050]

Transform the augmented matrix into row echelon form.

Perform the transformation R1R15.

[55105155505152005010101050][12310152005010101050]

Perform the transformation R2R215R1.

[123101515(1)2015(2)015(3)5015(10)10101050][123100104510010101050]

Perform the transformation R3R310R1.

[12310010451001010(1)1010(2)1010(3)5010(10)][12310010451000102050]

Perform the transformation R2R210

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