Question

1. The container of a breakfast cereal usually lists the number of calories and the amounts of protein, carbohydrate, and fat contained in one serving of the cereal. The amounts for two common cereals are given below. Suppose a mixture of these two cereals is to be prepared that contains exactly 295 calories, 9 g of protein, 48 g of carbohydrate, and 8 g of fat. a. Set up a vector equation for this problem. Include a statement of what the variables in your equation represent. b. Write an equivalent matrix equation, and then determine if the desired mixture of the two cereals can be prepared.

Answer 1

Complete Question:

The container of a breakfast cereal usually lists the number of calories and the amounts of protein, carbohydrate, and fat contained in one serving of the cereal. The amounts for two common cereals are given below. Suppose a mixture of these two cereals is to be prepared that contains exactly 295 calories, 9 g of protein, 48 g of carbohydrate, and 8 g of fat. a. Set up a vector equation for this problem. Include a statement of what the variables in your equation represent. b. Write an equivalent matrix equation, and then determine if the desired mixture of the two cereals can be prepared.

Nutrient         General mills cheerios        Quaker 100% Natural cereal

Calories                       110                                           130

Protein (g)                    4                                                3

Carbohydrate (g)         20                                             18

Fat (g)                           2                                                 5

Solution:

Representation of data in equation.

x1 is the number of servings of Cheerios and

x2 is the number of servings of Natural Cereal

110$x_{1}$ + 130$x_{2}$  = 295

4$x_{1}$ + 3$x_{2}$ = 9

20$x_{1}$ + 18$x_{2}$ = 48

2$x_{1}$ + 5$x_{2}$ = 8

Representation of data in vector equation,

$x_{1}$$\left[\begin{array}{ccc}110\\4\\20\\2\end{array}\right]$   + $x_{2}$ $\left[\begin{array}{ccc}130\\3\\18\\5\end{array}\right]$  = $\left[\begin{array}{ccc}295\\9\\48\\8\end{array}\right]$

Equivalent matrix equation of data

$\left[\begin{array}{ccc}110&130&295\\4&3&9\\20&18&48\\2&5&8\end{array}\right]$

Row reduce the matrix

$\left[\begin{array}{ccc}110&130&295\\4&3&9\\20&18&48\\2&5&8\end{array}\right]$  ≅ $\left[\begin{array}{ccc}1&0&1.5\\0&1&1\\0&0&0\\0&0&0\end{array}\right]$

1.5 serving of cherrios with 1 serving of natural cereal requried to make desired mixture.

$x_{1}$ = 1.5 , $x_{2}$ = 1

a) $x_{1}$ $\left[\begin{array}{ccc}110\\4\\20\\2\end{array}\right]$ + $x_{2}$ $\left[\begin{array}{ccc}130\\3\\18\\5\end{array}\right]$  =  $\left[\begin{array}{ccc}295\\9\\48\\8\end{array}\right]$

b) $x_{1}$ = 1.5 , $x_{2}$ = 1