Question

(1) The table below shows the calories, fat, and carbohydrates per ounce for three brands of cereal, and the total amount of calories, fat, and carbohydrates required for a special diet. Calories Fat Carbohydrates Cereal Brand X 50 0 22 Cereal Brand Y 108 0.1 25.5 Cereal Brand Z 127 5.5 18 Total Required 393 5.7 91 (i) Using the information in the table above, write a system of three equations. (ii) Write a system in matrix form Ax b  . (ii) Using the Cramer's Rule or Inverse Method, find the amount of each brand of cereal that will give the level of nutrition required. 

Answer 1

Using cramer's Rule;

$\begin{pmatrix} 50&108& 127\\ 0&0.1 & 5.5\\ 22&25.5&18 \end{pmatrix}\begin{pmatrix} x \\ y \\ z \end{pmatrix}=\begin{pmatrix} 393 \\ 5.7\\ 91 \end{pmatrix}$

Let $D=$ $\begin{pmatrix} 50&108 & 127 \\ 0 & 0.1&5.5\\ 22&25.5&18 \end{pmatrix}$

$D_x=\begin{pmatrix} 393& 108&127\\ 5.7&0.1 & 5.5\\ 91&25.5&18 \end{pmatrix}$

$D_y=\begin{pmatrix} 50&393 & 127 \\ 0 & 5.7&5.5\\ 22&91&18 \end{pmatrix}$

$D_2= \begin{pmatrix} 50&108 & 393 \\ 0&0.1 & 5.7\\ 22&25.5&91 \end{pmatrix}$

Det $D=50\begin{vmatrix} 0.1 & 5.5 \\ 25.5 & 18 \end{vmatrix}-108\begin{vmatrix} 0& 5.5 \\ 22 & 18 \end{vmatrix}+127\begin{vmatrix} 0 & 0.1\\ 22 & 25.5 \end{vmatrix}$

$=50(-138.45)-108(-121)+127(-2.2)$

$=5866.1$

Det $D_x=393\begin{vmatrix} 0.1& 5.5 \\ 25.5& 18 \end{vmatrix}-108\begin{vmatrix} 5.7& 5.5 \\ 91& 18 \end{vmatrix}+127\begin{vmatrix} 5.7 & 0.1\\ 91 & 25.5 \end{vmatrix}$

$=393(-138.45)-108(-397.9)+127(136.25)$

$=5866.1$

Det $D_y=50\begin{vmatrix} 5.7 & 5.5 \\ 91& 18 \end{vmatrix}-393\begin{vmatrix} 0& 5.5 \\ 22& 18 \end{vmatrix}+127\begin{vmatrix} 0& 5.7 \\ 22 & 91 \end{vmatrix}$

$=50(-397.9)-393(-121)+127(-125.4)$

$=11732.2$

Det $D_2=50\begin{vmatrix} 0.1 & 5.7 \\ 25.5 & 91 \end{vmatrix}-108\begin{vmatrix} 0& 5.7 \\ 22 & 91 \end{vmatrix}+393\begin{vmatrix} 0& 0.1\\ 22& 25.5 \end{vmatrix}$

$=50(-136.25)-108(-125.4)+393(-22)$

$=5866.1$

$x=\frac{|D_x|}{|D|}=\frac{5866.1}{5866.1}=1,$

$y=\frac{|D_y|}{|D|}=\frac{11732.2}{5866.1}=2$

$z=\frac{|D_z|}{|D|}=\frac{5866.1}{5866.1}=1$

Amount of each brand of cereal that will give the level of nutrition required

Cereal Brand $X=1$ ounce

Cereal Brand $Y=2$ ounces

Cereal Brand $Z=1$ ounce