Chapter 10.3, Problem 43E

LIFE SCIENCE APPLICATIONS

Dietetics In Exercise 39 from section 10.2, label the matrices

$\left[\begin{array}{cccc}2& 1& 2& 1\\ 3& 2& 2& 1\\ 4& 3& 2& 1\end{array}\right],\left[\begin{array}{ccc}5& 0& 7\\ 0& 10& 1\\ 0& 15& 2\\ 10& 12& 8\end{array}\right]\text{and}\left[\begin{array}{c}8\\ 4\\ 5\end{array}\right]$

found in parts a, b, and c, respectively, $X$, $Y$, and $Z$.

a. Find the product matrix $XY$. What do the entries of this matrix represent?

b. Find the product matrix $YZ$. What do the entries represent?

c. Find the products $\left(XY\right)Z$ and $X\left(YZ\right)$ and verify that they are equal. What do the entries represent?

LIFE SCIENCE APPLICATIONS

39. Dietetics A dietician prepares a diet specifying the amounts a patient should eat of four basic food groups: group 1, meats: group II, fruits and vegetables: group Ill, breads and starches: group IV, milk products. Amounts are given in “exchanges” that represent I oz (meat), 1/2 cup (fruits and vegetables). 1 slice (bread), 8 oz (milk), or other suitable measurements.

a. The number of “exchanges” for breakfast for each of the tour food groups, respectively, are $2,1,2$ and $1$; for lunch, $3,2,2$ and $1$; and for dinner, $4,3,2$ and $1$. Write a $3\times 4$ matrix using this information.

b. The amounts of fat, carbohydrates, and protein (in appropriate units) in each food group, respectively, are as follows.

Fat: $5,0,0,10$

Carbohydrates: $0,10,15,12$

Protein: $7,1,2,8$

Use this information to write a $4\times 3$ matrix.

c. There are $8$ calories per exchange of fat, $4$ calories per exchange of carbohydrates, and $5$ calories per exchange of protein. Summarize this data in a $3\times 1$ matrix.