Exercise 7

Suppose that a company has three plants, called Plants 1, 2, and 3, that produce milk $M$ and yogurt $Y$. For every hour of operation,

• Plant $1$ produces 20 units of milk and 15 units of yogurt.

• Plant $2$ produces 30 units of milk and 5 units of yogurt.

• Plant $3$ produces 0 units of milk and 40 units of yogurt.

1. Suppose that $x_1$, $x_2$, and $x_3$ record the amounts of time that the three plants are operated and that $M$ and $Y$ record the amount of milk and yogurt produced. If we write $\xvec=\threevec{x_1}{x_2}{x_3}$ and $\yvec = \twovec{M}{Y}$, find the matrix $A$ that defines the matrix transformation $T(\xvec) = \yvec$.

2. Furthermore, suppose that producing each unit of

   • milk requires 5 units of electricity and 8 units of labor.

   • yogurt requires 6 units of electricity and 10 units of labor.

   If we write the vector $\zvec = \twovec{E}{L}$ to record the required amounts of electricity $E$ and labor $L$, find the matrix $B$ that defines the matrix transformation $S(\yvec) = \zvec$.

3. If $\xvec = \threevec{30}{20}{10}$ describes the amounts of time that the three plants are operated, how much milk and yogurt is produced? How much electricity and labor are required?

4. Find the matrix $C$ that describes the matrix transformation $R(\xvec)=\zvec$ that gives the required amounts of electricity and labor when the each plants is operated an amount of time given by the vector $\xvec$.