A local bakery sells two types of muffins: blueberry and bran. Store 1 sells 30 blueberry … — ACT

A local bakery sells two types of muffins: blueberry and bran. Store 1 sells 30 blueberry muffins and 20 bran muffins. Store 2 sells 40 blueberry muffins and 35 bran muffins. Blueberry muffins cost $3 each, and bran muffins cost$ 2 each. This information is represented by the inventory matrix $M$ and the price matrix $P$ :

M = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix}

P = \begin{pmatrix} 3 \\ 2 \end{pmatrix}

Which of the following matrices represents the total revenue from muffin sales for Store 1 and Store 2, calculated using the matrix product $MP$ ?

A
$\begin{pmatrix} 90 & 40 \\ 120 & 70 \end{pmatrix}$
B
$\begin{pmatrix} 120 \\ 185 \end{pmatrix}$
$\begin{pmatrix} 130 \\ 190 \end{pmatrix}$ Answer
D
$\begin{pmatrix} 130 & 190 \end{pmatrix}$
E
$\begin{pmatrix} 150 \\ 210 \end{pmatrix}$

Answer

The column matrix with elements 130 and 190

The correct answer is the column matrix with elements 130 and 190. Multiplying the first row of the inventory matrix by the price column matrix yields the revenue for Store 1 (

30 \times 3 + 20 \times 2 = 130

). Multiplying the second row by the price column matrix yields the revenue for Store 2 (

40 \times 3 + 35 \times 2 = 190

). Since the product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 column matrix, the final result is correctly formatted.

Step-by-Step Solution

Set up the matrix multiplication of the 2 by 2 matrix and the 2 by 1 matrix.

MP = \begin{pmatrix} 30 & 20 \\ 40 & 35 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix}

To calculate total revenue, we multiply the quantity matrix by the price matrix.

Calculate the first element of the resulting matrix (Store 1 total revenue) by taking the dot product of the first row of the inventory matrix and the price matrix.

30 \times 3 + 20 \times 2 = 90 + 40 = 130

This pairs the quantity of each muffin type sold at Store 1 with its respective price.

Calculate the second element of the resulting matrix (Store 2 total revenue) by taking the dot product of the second row of the inventory matrix and the price matrix.

40 \times 3 + 35 \times 2 = 120 + 70 = 190

This pairs the quantity of each muffin type sold at Store 2 with its respective price.

Assemble the resulting products into a 2 by 1 column matrix.

\begin{pmatrix} 130 \\ 190 \end{pmatrix}

The product of a 2 by 2 matrix and a 2 by 1 matrix is a 2 by 1 matrix.

Key Concept

Matrix multiplication of a 2 by 2 matrix and a 2 by 1 matrix in a real-world context.

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