##### Exercise6

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 $$P$$ produces 20 units of milk and 15 units of yogurt.

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

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

1. 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.

2. 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 $$T(\xvec) = \yvec\text{.}$$

3. 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 $$Y\text{.}$$

4. 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 $$S(\yvec) = \zvec\text{.}$$

5. 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?

6. 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 $$\xvec\text{.}$$

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