Chapter 12.4, Problem 58PPSE

Solution

a.

To find : The weight of each ingredient that caterer uses if a caterer combines ingredients to make a paella, a Spanish fiesta dish by solving the system of equation formed.

The system of three equations to find the weight of each ingredient is $\left[\begin{array}{ccc}1& 1& 1\\ 1.50& 0.4& 6\\ 100& 20& 50\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}18\\ 29.50\\ 850\end{array}\right]$ .

Given information :

Caterer uses if a caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 18 lb, costs $\$29.50$ and supplies 850 g of protein.

Explanation :

Take the caterer requirement as $x$ lb of chicken, $y$ lb of rice and $z$ lb of shellfish for preparing a Spanish dish paella.

The paella weighs 18 lb.

So,

  $x+y+z=18$

Write the equations for its cost and protein supply.

  $\begin{array}{c}1.5x+0.4y+6z=29.50\\ 100x+20y+50z=850\end{array}$

Write the system of equation in matrix form.

  $\left[\begin{array}{ccc}1& 1& 1\\ 1.50& 0.4& 6\\ 100& 20& 50\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}18\\ 29.50\\ 850\end{array}\right]$

b.

To calculate: The pounds of ingredients to be used if caterer uses if a caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 18 lb, costs $\$29.50$ and supplies 850 g of protein.

The caterer used 5 lb of chicken, 10 lb of rice and 3 lb of shellfish.

Given information:

Caterer uses if a caterer combines ingredients to make a paella, a Spanish fiesta dish. The paella weighs 18 lb, costs $\$29.50$ and supplies 850 g of protein.

Formula Used:

Solve the matrix to find the values of each ingredient.

Calculation :

Consider the following matrix.

  $\left[\begin{array}{ccc}1& 1& 1\\ 1.50& 0.4& 6\\ 100& 20& 50\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}18\\ 29.50\\ 850\end{array}\right]$

Solve the matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 1.50& 0.4& 6\\ 100& 20& 50\end{array}\right]$ using graphing calculator.

Enter matrix $A=\left[\begin{array}{ccc}1& 1& 1\\ 1.50& 0.4& 6\\ 100& 20& 50\end{array}\right]$ into graphing calculator.

Select the $\text{EDIT}$ option of the matrix feature to edit the matrix $\left[\text{A}\right]$ . Specify a $3\times 3$ matrix by pressing $3\text{ enter 3 enter}$ .

Now, enter the matrix element one row at a time, pressing enter after each element.

Then use the quit feature to return to the main screen.

  

  

To compute the operation $X=A^{-1}B$ , use $\text{NAMES}$ option of the matrix feature to select each matrix.

  

  

Press $\text{MATRIX}$ , the number of matrix 1, and enter then press $X^{-1}$ key and press the number of matrix 2 and then $\text{ENTER}$ to see the values.

  

Thus, the caterer used 5 lb of chicken, 10 lb of rice and 3 lb of shellfish.