Chapter 2.3, Problem 76E

Bakery Sales A bakery makes three types of cookies, I, II, and III. Each type of cookie is made from the four ingredients A, B, C, and D. The number of units of each ingredient used in each type of cookie is given by the matrix M. The cost per unit of each of the four ingredients (in cents) is given by the matrix N. The selling price for each of the cookies (in cents) is given by the matrix S. The baker receives an order for 10 type I cookies, 20 type II cookies, and 15 type III cookies, as represented by the matrix R.

$\begin{array}{l}\begin{array}{cccc}A& B& C& D\end{array}\text{Cost}\\ M\left[\begin{array}{cccc}1& 0& 2& 4\\ 3& 2& 1& 1\\ 2& 5& 3& 1\end{array}\right]\begin{array}{c}\text{I}\\ \text{II}\\ \text{III}\end{array}N=\left[\begin{array}{c}10\\ 20\\ 15\\ 17\end{array}\right]\begin{array}{c}\text{A}\\ \text{B}\\ \text{C}\\ \text{D}\end{array}\end{array}$

$\begin{array}{l}\text{Selling}\\ \text{price}\\ S=\left[\begin{array}{c}175\\ 150\\ 225\end{array}\right]\begin{array}{c}\text{I}\\ \text{II}\\ \text{III}\end{array}\end{array}$ $\begin{array}{l}\begin{array}{ccc}\text{I}& \text{II}& \text{III}\end{array}\\ R=\left[\begin{array}{ccc}10& 20& 15\end{array}\right]\text{Order}\end{array}$

Calculate and interpret the following:

(a) $RM$

(b) $MN$

(c) $RMN$

(d) $S-MN$

(e) $R\left(S-MN\right)$

(f) $RS$