Chapter 5.7, Problem 86E

To determine

(a)

To find:

Using matrix multiplication to find a matrix giving the comparative cost per bakery item for the two purchase options

The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A.

$\begin{array}{l}\begin{array}{ccccc}\text{Eggs}& \text{Flour}& \text{Sugar}& \text{Shortening}& \text{Milk}\end{array}\\ \begin{array}{c}\text{Rolls (doz)}\\ \text{Bread (loaf)}\\ \text{Cake}\\ \text{Pie (crust)}\end{array}\left[\begin{array}{ccccc}1& 4& \frac{1}{4}& \frac{1}{4}& 1\\ 0& 3& 0& \frac{1}{4}& 0\\ 4& 3& 2& 1& 1\\ 0& 1& 0& \frac{1}{3}& 0\end{array}\right]=A\end{array}$

The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix B.

$\begin{array}{l}\text{ Cost}\\ \begin{array}{cc}\text{ Large Lot}& \text{Small Lot}\end{array}\\ \begin{array}{c}\text{Eggs}\\ \text{Flour}\\ \text{Sugar}\\ \text{Shortening}\\ \text{Milk}\end{array}\left[\begin{array}{cc}5& 5\\ 8& 10\\ 10& 12\\ 12& 15\\ 5& 6\end{array}\right]=B\end{array}$

To determine

(b)

To find:

Suppose a day’s orders consist of 20 dozen sweet rolls, 200 loaves of bread, 50 cakes, and 60 pies. Write the orders as a $1\times 4$ matrix, and, using matrix multiplication, to write as a matrix the amount of each ingredient needed to fill the day’s orders

The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A.

$\begin{array}{l}\begin{array}{ccccc}\text{Eggs}& \text{Flour}& \text{Sugar}& \text{Shortening}& \text{Milk}\end{array}\\ \begin{array}{c}\text{Rolls (doz)}\\ \text{Bread (loaf)}\\ \text{Cake}\\ \text{Pie (crust)}\end{array}\left[\begin{array}{ccccc}1& 4& \frac{1}{4}& \frac{1}{4}& 1\\ 0& 3& 0& \frac{1}{4}& 0\\ 4& 3& 2& 1& 1\\ 0& 1& 0& \frac{1}{3}& 0\end{array}\right]=A\end{array}$

The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix B.

$\begin{array}{l}\text{ Cost}\\ \begin{array}{cc}\text{ Large Lot}& \text{Small Lot}\end{array}\\ \begin{array}{c}\text{Eggs}\\ \text{Flour}\\ \text{Sugar}\\ \text{Shortening}\\ \text{Milk}\end{array}\left[\begin{array}{cc}5& 5\\ 8& 10\\ 10& 12\\ 12& 15\\ 5& 6\end{array}\right]=B\end{array}$

To determine

(c)

To find:

Using matrix multiplication to find a matrix giving the costs under the two purchase options to fill the day’s orders.

The Bread Box, a small neighborhood bakery, sells four main items: sweet rolls, bread, cakes, and pies. The amount of each ingredient (in cups, except for eggs) required for these items is given by matrix A.

$\begin{array}{l}\begin{array}{ccccc}\text{Eggs}& \text{Flour}& \text{Sugar}& \text{Shortening}& \text{Milk}\end{array}\\ \begin{array}{c}\text{Rolls (doz)}\\ \text{Bread (loaf)}\\ \text{Cake}\\ \text{Pie (crust)}\end{array}\left[\begin{array}{ccccc}1& 4& \frac{1}{4}& \frac{1}{4}& 1\\ 0& 3& 0& \frac{1}{4}& 0\\ 4& 3& 2& 1& 1\\ 0& 1& 0& \frac{1}{3}& 0\end{array}\right]=A\end{array}$

The cost (in cents) for each ingredient when purchased in large lots or small lots is given by matrix B.

$\begin{array}{l}\text{ Cost}\\ \begin{array}{cc}\text{ Large Lot}& \text{Small Lot}\end{array}\\ \begin{array}{c}\text{Eggs}\\ \text{Flour}\\ \text{Sugar}\\ \text{Shortening}\\ \text{Milk}\end{array}\left[\begin{array}{cc}5& 5\\ 8& 10\\ 10& 12\\ 12& 15\\ 5& 6\end{array}\right]=B\end{array}$