Chapter 2.5, Problem 13CFU

To determine

To find: the amount of each metal that should be used by a metallurgist.

Answer to Problem 13CFU

  $\begin{array}{l}x=8\\ y=12\end{array}$

Explanation of Solution

Given information:

Let the two type of metal be x and y respectively.

20 kg of aluminium alloy is made of x and y. 55% of x and 80% of y makes up 70% of $\left(x+y\right)$ .

Calculation:

From the data, the equations can be setup as follows:

  $x+y=20$ ...(1)

  $0.5x+0.80y=0.70\left(x+y\right)$...(2)

Solve the second equation and simplify as follows:

  $\begin{array}{c}0.5x+0.80y=0.70x+0.70y\\ 0.70x-0.5x=0.80y-0.70y\\ 0.15x=0.10y\\ 15x-10y=0 \mathrm{...}\left(3\right)\end{array}$

Set up the matrix for the equations (1) and (3) as follows:

  $\begin{array}{l}\begin{array}{ccc}\text{ A }\cdot & B& =\text{C}\end{array}\\ \left[\begin{array}{cc}\begin{array}{l}1\\ 15\end{array}& \begin{array}{l}4\\ -5\end{array}\end{array}\right]\cdot \left[\begin{array}{l}x\\ y\end{array}\right]=\left[\begin{array}{l}20\\ 0\end{array}\right]\end{array}$

Evaluate the determinant of $\left|\begin{array}{cc}\begin{array}{l}1\\ 15\end{array}& \begin{array}{l}4\\ -5\end{array}\end{array}\right|$ as shown below.

  $\begin{array}{c}\det =1\left(-15\right)-4\left(-5\right)\\ =-25\end{array}$

Evaluate the inverse of the coefficient matrix as shown below.

  $A^{-1}=\frac{1}{-25}\left[\begin{array}{cc}\begin{array}{l}-10\\ -15\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]$

Now substitute value in the given equation as follows:

  $\begin{array}{c}{A}^{-1}B=X\\ \frac{1}{-25}\left[\begin{array}{cc}\begin{array}{l}-10\\ -15\end{array}& \begin{array}{l}-1\\ 1\end{array}\end{array}\right]\cdot \left[\begin{array}{l}20\\ 0\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\\ \left[\begin{array}{l}8\\ 12\end{array}\right]=\left[\begin{array}{l}x\\ y\end{array}\right]\end{array}$

Thus, the value of x and y is $8$ and $12$ respectively.