A local farm sells organic fruit baskets at a weekend market. The number of small and larg… — ACT

A local farm sells organic fruit baskets at a weekend market. The number of small and large baskets sold on Saturday and Sunday is represented by matrix $Q$ , where row 1 represents Saturday, row 2 represents Sunday, column 1 represents small baskets, and column 2 represents large baskets:

Q = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix}

The selling price and the production cost, in dollars, for each type of basket are represented by matrix $P$ , where row 1 represents small baskets, row 2 represents large baskets, column 1 represents the selling price, and column 2 represents the production cost:

P = \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

The product matrix $R = QP$ represents the total revenue and total production cost for each day. Which of the following matrices represents $R$ ?

$\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}$ Answer
B
$\begin{bmatrix} 600 & 240 \\ 500 & 600 \end{bmatrix}$
C
$\begin{bmatrix} 760 & 850 \\ 1240 & 1350 \end{bmatrix}$
D
$\begin{bmatrix} 1350 & 1550 \\ 680 & 760 \end{bmatrix}$
E
$\begin{bmatrix} 1450 & 680 \\ 1550 & 660 \end{bmatrix}$

Answer

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

The correct product matrix is obtained by performing matrix multiplication

QP

. Multiplying the first row of

Q

by the first column of

P

yields the Saturday revenue:

40 \times 15 + 30 \times 25 = 1350

. Multiplying the first row of

Q

by the second column of

P

yields the Saturday cost:

40 \times 8 + 30 \times 12 = 680

. Repeating this for the second row of

Q

(Sunday) yields the Sunday revenue of

1550

and Sunday cost of

760

. Thus, the resulting matrix is the one with row 1 equal to

[1350, 680]

and row 2 equal to

[1550, 760]

Step-by-Step Solution

Set up the matrix multiplication product

R = QP

R = \begin{bmatrix} 40 & 30 \\ 20 & 50 \end{bmatrix} \begin{bmatrix} 15 & 8 \\ 25 & 12 \end{bmatrix}

To find the total revenue and production cost for Saturday and Sunday, we must multiply the quantity matrix by the price-cost matrix.

Calculate the elements of the first row of the product matrix

R

, representing Saturday's revenue and cost.

Row 1, Column 1 (Saturday Revenue):

40(15) + 30(25) = 600 + 750 = 1350

. Row 1, Column 2 (Saturday Cost):

40(8) + 30(12) = 320 + 360 = 680

Multiply the first row of matrix

Q

by the columns of matrix

P

to determine Saturday's financial values.

Calculate the elements of the second row of the product matrix

R

, representing Sunday's revenue and cost.

Row 2, Column 1 (Sunday Revenue):

20(15) + 50(25) = 300 + 1250 = 1550

. Row 2, Column 2 (Sunday Cost):

20(8) + 50(12) = 160 + 600 = 760

Multiply the second row of matrix

Q

by the columns of matrix

P

to determine Sunday's financial values.

Combine the calculated elements into the final

2 \times 2

matrix.

\begin{bmatrix} 1350 & 680 \\ 1550 & 760 \end{bmatrix}

Placing the values in their corresponding row and column positions yields the complete product matrix.

Key Concept

Matrix Multiplication in Applied Word Problems

Estimated Time:1m 30s

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