1.3 Applications of Linear Systems

1. A chef is trying to create a unique pepper blend for a new restaurant…

The intended recipe will be 70% black peppercorns and 30% white peppercorns. The chef has three different pepper blends on hand. The ratio of black to white peppercorns of the three blends are shown in the table below.

	Black	White
Blend A	90	10
Blend B	50	50
Blend C	35	65
Desired	70	30

$$\begin{array}{r}0.7=0.9A+0.5B+0.35C\\ 0.3=0.1A+0.5B+0.65C\end{array}$$ $$\left[\begin{array}{ccccc}0.9& 0.5& 0.35& |& 0.7\\ 0.1& 0.5& 0.65& |& 0.3\\ 1& 1& 1& |& 1\end{array}\right]$$

RREF:

$$\left[\begin{array}{ccccc}1& 0& -3/8& |& 1/2\\ 0& 1& 11/8& |& 1/2\\ 0& 0& 0& |& 0\end{array}\right]$$

Answer:
Honestly not sure how to finish this using given methods but I noticed that mixing even parts A and B gives the desired mixture…

2. Consider the wire grid shown below…

The temperature at the black nodes is held constant. When the temperature is at equilibrium, the temperature at the white nodes is the average of the temperatures of the adjacent nodes. Find the equilibrium temperature of the wire grid.

$$T_1=1/4(30+10+T_3+T_2)$$ $$4T_1-T_2-T_3=40$$ $$T_2=1/4(70+T_3+T+1+30)$$ $$-T_1+4T_2-T_3=100$$ $$T_3=1/4(10+T_1+T_2+70)$$ $$T_1+T_2-4T_3=-80$$ $$\left[\begin{array}{ccccc}4& -1& -1& |& 40\\ -1& 4& -1& |& 100\\ 1& 1& -4& |& -80\end{array}\right]$$

RREF:

$$\left[\begin{array}{ccccc}1& 0& 0& |& 30\\ 0& 1& 0& |& 42\\ 0& 0& 1& |& 38\end{array}\right]$$

Answer:
$T_1=30,T_2=42,T_3=48$