A local bakery sells chocolate chip cookies for 1.50 each and oatmeal raisin cookies for 1… — ACT

A local bakery sells chocolate chip cookies for $1.50 each and oatmeal raisin cookies for$ 1.25 each. The table below shows the number of cookies sold on Monday and Tuesday:

Day	Chocolate Chip	Oatmeal Raisin
Monday	$80$	$60$
Tuesday	$95$	$70$

If the sales data is represented by the matrix $S = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix}$ and the prices are represented by the matrix $P = \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}$ , which of the following matrices represents the total revenue, in dollars, generated on Monday and Tuesday, respectively?

A
$\begin{bmatrix} 190.00 \\ 223.75 \end{bmatrix}$
B
$\begin{bmatrix} 120.00 & 75.00 \\ 142.50 & 87.50 \end{bmatrix}$
$\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}$ Answer
D
$\begin{bmatrix} 238.75 \\ 177.50 \end{bmatrix}$
E
$\begin{bmatrix} 185.00 \\ 220.00 \end{bmatrix}$

Answer

The matrix representing the revenue is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

The correct matrix is obtained by performing standard matrix multiplication of

S

(dimensions

2 \times 2

) and

P

(dimensions

2 \times 1

). The resulting matrix has dimensions

2 \times 1

. The first element is the sum of the products of the first row of

S

and the column of

P

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. The second element is the sum of the products of the second row of

S

and the column of

P

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

. This gives the matrix containing the values 195.00 and 230.00.

Step-by-Step Solution

Set up the matrix multiplication of sales and prices

Multiply the

2 \times 2

sales matrix

S

by the

2 \times 1

price matrix

P

S \times P = \begin{bmatrix} 80 & 60 \\ 95 & 70 \end{bmatrix} \begin{bmatrix} 1.50 \\ 1.25 \end{bmatrix}

Matrix multiplication aligns the quantities sold of each item with their respective unit prices to calculate the total revenue per day.

Perform the matrix multiplication for each row

Monday's revenue is

80 \times 1.50 + 60 \times 1.25 = 120 + 75 = 195

. Tuesday's revenue is

95 \times 1.50 + 70 \times 1.25 = 142.50 + 87.50 = 230

Multiplying the elements of each row in the first matrix by the corresponding elements in the column of the second matrix, and then summing them, calculates the dot product for each day.

Write the results in a

2 \times 1

matrix

The final product matrix is

\begin{bmatrix} 195.00 \\ 230.00 \end{bmatrix}

The product of a

2 \times 2

matrix and a

2 \times 1

matrix is a

2 \times 1

matrix containing the total values for each row.

Key Concept

Matrix Multiplication in Word Problems

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