Exercise 6
Suppose that a company has three plants, called Plants 1, 2, and 3, that produce milk \(M\) and yogurt \(Y\text{.}\) 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.
Suppose that \(x_1\text{,}\) \(x_2\text{,}\) and \(x_3\) record the amounts of time that the three plants are operated. Find expressions for the number of units of milk \(M\) and yogurt \(Y\) produced.
If we write \(\xvec=\threevec{x_1}{x_2}{x_3}\) and \(\yvec = \twovec{M}{Y}\text{,}\) find the matrix \(A\) that defines the matrix transformation \(T(\xvec) = \yvec\text{.}\)

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.
Write expressions for the required amounts of electricity \(E\) and labor \(L\) in terms of \(M\) and \(Y\text{.}\)
If we write the vector \(\zvec = \twovec{E}{L}\) to record the required amounts of electricity and labor, find the matrix \(B\) that defines the matrix transformation \(S(\yvec) = \zvec\text{.}\)
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?
Find the matrix \(C\) that describes the matrix transformation \(R(\xvec)=\zvec\) that gives the required amounts of electricity and labor when the plants are operated times given by vector \(\xvec\text{.}\)