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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 Refer to exercise 61 . Suppose food type C has been improperly labeled, and it actually contains 4 mg of folic acid, 6 mg choline, and 5 mg of inositol per ounce. Would it still be possible to use matrix inversion to solve parts (b), (c), (d) of exercise 61 ? Why or why not?

6 1 . Nutrition A nutritionist is studying the effects of the nutrients folic acid, choline, and inositol. He has three types of food available, and each type contains the following amounts of these nutrients per ounce.

Type A Type B Type C
Folic acid (mg) 3 1 3
Choline (mg) 4 2 4
Inositol (mg) 3 2 4

(a) Find the inverse of the matrix

[ 3 1 3 4 2 4 3 2 4 ]

and use it to solve the remaining parts of this problem.

(b) How many ounces of each food should the nutritionist feed his laboratory rats if he wants their daily diet to contain 10 mg of folic acid, 14 mg of choline, and 13 mg of inositol?

(c) How much of each food is needed to supply 9 mg of folic acid, 12 mg of choline, and 10 mg of inositol?

(d) Will any combination of these foods supply 2mg of folic acid, 4 mg of choline, and 11 mg of inositol?

To determine

(a)

To find:

The inverse of the matrix after changing the food type C.

Explanation

Given:

Food type C has been improperly labeled, and it actually contains 4 mg of folic acid, 6 mg choline, and 5 mg of inositol per ounce.

Approach:

Change the Food type C given in table of question 61 with the 4 mg of folic acid, 6 mg choline, and 5 mg of inositol per ounce. New table which shows the nutrients folic acid, choline, and inositol for three types of food is shown below.

Type A Type B Type C
Folic acid (mg) 3 1 4
Choline (mg) 4 2 6
Inositol (mg) 3 2 5

For finding the inverse use the following step.

Take a 3×6 matrix with left half as matrix B and right half as I3 and transform the left half into identity matrix by performing elementary row operations on the entire matrix, the right half becomes inverse of the matrix B.

Calculation:

Consider the matrix A=[314426325]

Take a matrix of dimension 3×6 with left half as matrix B and right half as I3 and perform row operations to transform the left half into I3.

[ 314100426010325001 ]R113R1[ 113431300426010325001 ]R2R24R1[ 113431300023234310325001 ]

To determine

(b)

To find:

The ounces of each food the nutritionist should feed his laboratory rats in order to keep their daily diet to contain 10 mg of folic acid, 14 mg of choline, and 13 mg of inositol.

To determine

(c)

To find:

The ounces of each food the nutritionist should feed his laboratory rats in order to keep their daily diet to contain 9 mg of folic acid, 12 mg of choline, and 10 mg of inositol.

To determine

(d)

To find:

The ounces of each food the nutritionist should feed his laboratory rats in order to keep their daily diet to contain 2 mg of folic acid, 4 mg of choline, and 11 mg of inositol.

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