Question

1. a children’s toy store sells mama, papa, and baby bears, each in small, medium and large sizes. during one week, the store sold 10 small - sized, 8 medium - sized, and 20 large - sized mama bears. it also sold 6 small - sized, 10 medium - sized, and 8 large - sized papa bears and 14 small - sized, 20 medium - sized and 14 large - sized baby bears.

a. express this information in a sales matrix s.
b. because this was a special sales week, the store expects to sell 50% fewer of each type of bear next week. how many of each medium - sized bear would it expect to sell? in terms of s, what matrix gives the amount of expected sales for each type of bear? write this matrix.

Explanation:

Part (a)

Step1: Define the matrix structure

Rows: Sizes (Small, Medium, Large)
Columns: Bear Types (Mama, Papa, Baby)

Step2: Fill in the values

• Small (S): Mama = 10, Papa = 6, Baby = 14
• Medium (M): Mama = 8, Papa = 10, Baby = 20
• Large (L): Mama = 20, Papa = 8, Baby = 14

So the sales matrix \( S \) is:
\[
S =

$$\begin{bmatrix} 10 & 6 & 14 \\ 8 & 10 & 20 \\ 20 & 8 & 14 \end{bmatrix}$$

\]

Part (b)

Step1: Determine the scaling factor

Since the store expects to sell 50% fewer, the scaling factor is \( 1 - 0.5 = 0.5 \) (or 50% of the original sales).

Step2: Find the expected sales matrix

The expected sales matrix \( E \) is \( 0.5 \times S \) (scalar multiplication of matrix \( S \) by 0.5).

Step3: Calculate medium - sized bears

• Medium - sized Mama Bear: \( 0.5\times8 = 4 \)
• Medium - sized Papa Bear: \( 0.5\times10 = 5 \)
• Medium - sized Baby Bear: \( 0.5\times20 = 10 \)

Step4: Write the expected sales matrix

\[
E=0.5\times

$$\begin{bmatrix} 10 & 6 & 14 \\ 8 & 10 & 20 \\ 20 & 8 & 14 \end{bmatrix}$$

=

$$\begin{bmatrix} 0.5\times10&0.5\times6&0.5\times14\\ 0.5\times8&0.5\times10&0.5\times20\\ 0.5\times20&0.5\times8&0.5\times14 \end{bmatrix}$$

=

$$\begin{bmatrix} 5&3&7\\ 4&5&10\\ 10&4&7 \end{bmatrix}$$

\]

Answer:

Part (a)

The sales matrix \( S \) is \(\boldsymbol{

$$\begin{bmatrix}10 & 6 & 14 \\8 & 10 & 20 \\20 & 8 & 14\end{bmatrix}$$

}\)

Part (b)

• Expected medium - sized bears: 4 (Mama), 5 (Papa), 10 (Baby)
• Expected sales matrix: \(\boldsymbol{

$$\begin{bmatrix}5&3&7\\4&5&10\\10&4&7\end{bmatrix}$$

}\) (which is \( 0.5S \))