Suppose that we work for a company that produces baked goods, including cakes, donuts, and eclairs. Our company operates two plants, called Plant 1 and Plant 2. In one hour of operation,

  1. If plant 1 operates for \(x_1\) hours and Plant 2 for \(x_2\) hours, how many cakes \(C\) does the company produce? How many donuts \(D\text{?}\) How many eclairs \(E\text{?}\)

  2. We define a matrix transformation \(T(\xvec) = \threevec{C}{D}{E}\) where \(\threevec{C}{D}{E}\) represents the number of baked goods produced when the plants are operated for times \(\xvec=\twovec{x_1}{x_2}\text{.}\) If \(T(\xvec) = A\xvec\text{,}\) what are the dimensions of the matrix \(A\text{?}\)

  3. Find the vector \(T\left(\twovec{1}{0}\right)\) and the vector \(T\left(\twovec{0}{1}\right)\) and use your results to write the matrix \(A\text{.}\)

  4. If we operate Plant 1 for 40 hours and Plant 2 for 50 hours, how many baked goods have we produced?

  5. Suppose the marketing department says we need to produce 1500 cakes, 4700 donuts, and 3300 eclairs. Is it possible to meet this order? If so, how long should the two plants operate?

  6. Let's now consider the needed ingredients:

    • Each cake requires 4 units of flour and and 2 units of sugar.

    • Each donut requires 1 unit of flour and 1 unit of sugar.

    • Each eclair requires 1 units of flour and 2 units of sugar.

    Suppose we make \(C\) cakes, \(D\) donuts, and \(E\) eclairs. How many units of flour \(F\) are required? How many units of sugar \(S\text{?}\)

  7. Write a matrix \(B\) that defines the matrix transformation \(R\left(\threevec{C}{D}{E}\right) = \twovec{F}{S}\text{.}\)

  8. If Plant 1 operates for 30 hours and Plant 2 operates for 20 hours, how many units of flour and sugar are required?

  9. We can consider the matrix transformation \(P(\xvec) = \twovec{F}{S}\) that tells us how many units of flour and sugar are required when we operate the plants for \(x_1\) and \(x_2\) hours. Find the matrix that defines the transformation \(P\text{.}\)