A retail electronics store sells two models of wireless headphones: Solo, which sells for … — ACT

A retail electronics store sells two models of wireless headphones: Solo, which sells for $80, and Studio, which sells for$ 150. During a holiday sale, the number of headphones sold at the Downtown and Uptown store locations is represented by the matrix $N$ below, where the first row represents the Downtown store, the second row represents the Uptown store, the first column represents Solo headphones, and the second column represents Studio headphones:

N = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix}

The prices of the headphones are represented by the matrix $P$ below:

P = \begin{bmatrix} 80 \\ 150 \end{bmatrix}

Which of the following matrices represents the total revenue, in dollars, generated from headphone sales at each of the two stores, where the first row represents the Downtown store and the second row represents the Uptown store?

A
$\begin{bmatrix} 8,000 \\ 7,300 \end{bmatrix}$
B
$\begin{bmatrix} 5,600 \\ 9,000 \end{bmatrix}$
$\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}$ Answer
D
$\begin{bmatrix} 6,850 \\ 7,650 \end{bmatrix}$
E
$\begin{bmatrix} 3,200 & 3,750 \\ 2,400 & 5,250 \end{bmatrix}$

Answer

The column matrix with first row 6,950 and second row 7,650

The total revenue for each store is found by calculating the product of the sales matrix

N

and the price matrix

P

. This multiplication is performed by taking the dot product of each row in

N

with the column in

P

. For the first row (Downtown store), the calculation is

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

. For the second row (Uptown store), the calculation is

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

. Combining these results yields the column matrix containing 6,950 and 7,650.

Step-by-Step Solution

Set up the matrix multiplication to calculate the revenue, multiplying the sales matrix

N

by the price matrix

P

NP = \begin{bmatrix} 40 & 25 \\ 30 & 35 \end{bmatrix} \begin{bmatrix} 80 \\ 150 \end{bmatrix}

To find the total revenue for each store, the number of units sold (organized by store in rows and product type in columns) must be multiplied by the unit prices (organized by product type in rows).

Calculate the revenue for the Downtown store by taking the dot product of the first row of

N

and the column of

P

40 \times 80 + 25 \times 150 = 3,200 + 3,750 = 6,950

This combines the units sold of each headphone model at the Downtown store with their respective prices to find the store's total revenue.

Calculate the revenue for the Uptown store by taking the dot product of the second row of

N

and the column of

P

30 \times 80 + 35 \times 150 = 2,400 + 5,250 = 7,650

This combines the units sold of each headphone model at the Uptown store with their respective prices to find the store's total revenue.

Combine the results into the final product matrix.

\begin{bmatrix} 6,950 \\ 7,650 \end{bmatrix}

The resulting matrix contains the total revenues for the Downtown store in the first row and the Uptown store in the second row.

Key Concept

Matrix Multiplication and Modeling

Alternative Method

Instead of setting up the entire matrix multiplication, calculate the total revenue for the first store (Downtown) individually:

40 \times 80 + 25 \times 150 = 6,950

. This single calculation is enough to eliminate the options that do not have 6,950 in the first row.

Estimated Time:2m 0s

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