Q41. - Algebra 2

Question

WRITING IN MATH Answer the question that was posed at the beginning of the lesson.

 

In her job as a hospital dietician, Celeste designs weekly menus for her patients and tracks various nutrients for each daily diet. The table shows the Calories, proteins, and fat in a patient’s meals over a three-day period.

 

How can matrices be used to calculate daily dietary needs?

Include the following in your answer:

• Three matrices that represent breakfast, lunch, and dinner over the three-day period, and
• A matrix that represents the total Calories, protein, and fat consumed each day.

Step-by-Step Solution

Answer

The matrix representing the total Calories, protein, and fat consumed each day is:

$\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}\text{2608}& 80& 52\\ \text{2091}& 67& 82\\ \text{2620}& 65& 61\end{array}\right]$.

Step 1 - Define the concept of organizing data in matrix

A matrix is a rectangular array of variables or constants in horizontal rows and vertical columns, usually enclosed in brackets. Data is organized in such a way that each position has a purpose. Each value of the matrix is called an element.

Step 2 - Write the matrix depicting data for breakfast, lunch and dinner

The data in the table can be organized in three matrices representing breakfast, lunch and dinner. Let the first matrix for breakfast, second matrix for lunch and third matrix for dinner be A, B and C. Then, 

$\begin{array}{l}\text{ Breakfast}\\ A=\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}566& 18& 7\\ 482& 12& 17\\ 530& 10& 11\end{array}\right]\end{array}$     $\begin{array}{l}\text{ Lunch}\\ B=\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}785& 22& 19\\ 622& 23& 20\\ 710& 26& 12\end{array}\right]\end{array}$    and

$\begin{array}{l}\text{ Dinner}\\ C=\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}1257& 40& 26\\ 987& 32& 45\\ 1380& 29& 38\end{array}\right]\end{array}$

Step 3 - Add the matrices

In order to find a matrix that represents the total Calories, protein, and fat consumed each day add the three matrices A, B and C and simplify. 

$\begin{array}{c}\text{ Breakfast Lunch Dinner}\\ A+B+C=\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}566& 18& 7\\ 482& 12& 17\\ 530& 10& 11\end{array}\right]+\left[\begin{array}{ccc}785& 22& 19\\ 622& 23& 20\\ 710& 26& 12\end{array}\right]+\left[\begin{array}{ccc}1257& 40& 26\\ 987& 32& 45\\ 1380& 29& 38\end{array}\right]\\ =\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}566+785+1257& 18+22+40& 7+19+26\\ 482+622+987& 12+23+32& 17+20+45\\ 530+710+1380& 10+26+29& 11+12+38\end{array}\right]\\ =\begin{array}{c}\text{Day 1}\\ \text{Day 2}\\ \text{Day 3}\end{array}\left[\begin{array}{ccc}\text{2608}& 80& 52\\ \text{2091}& 67& 82\\ \text{2620}& 65& 61\end{array}\right]\end{array}$

In the matrix obtained, the first column gives the total calories consumed each day for three days, second column gives the total protein and third column gives the total fat consumed for three days.