Question

A online merchant, SchoolStuff sells pens for 40¢, pencils for 40¢ and erasers for 10¢. Office World sells erasers for 6¢ and Desks R Us sells erasers for 3¢. Find the least expensive place to order 500 pens, 300 pencils and 150 erasers.

Use matrix multiplication to find your answer. Show the matrices used (you may use a calculator for the calculations)

Answer 1

To find the least expensive place to order 500 pens, 300 pencils, and 150 erasers, we need to compare the total cost of purchasing these items from each merchant.

Let's set up a matrix multiplication problem to calculate the total cost. We'll use the following matrices:

A = [40 40 10] represents the cost of pens, pencils, and erasers from SchoolStuff respectively.

B = [500 300 150] represents the quantity of pens, pencils, and erasers required respectively.

C = [0.06 0.06 0.06] represents the cost of erasers from Office World.

D = [0.03 0.03 0.03] represents the cost of erasers from Desks R Us.

The matrix multiplication problem is:

AB^T + CD^T

where AB^T represents the cost of pens and pencils from SchoolStuff, and CD^T represents the cost of erasers from Office World and Desks R Us.

Calculating the matrices:

AB^T = [40 40 10] * [500 300 150]^T

= [40*500 + 40*300 + 10*150]

= [20000 + 12000 + 1500]

= [33500]

CD^T = [0.06 0.06 0.06] * [150]

= [0.06*150]

= [9]

Adding the results:

AB^T + CD^T = [33500] + [9]

= [33509]

Therefore, the total cost of purchasing 500 pens, 300 pencils, and 150 erasers from SchoolStuff, Office World, and Desks R Us respectively is 33509 cents.

Since SchoolStuff has the lowest total cost, it is the least expensive place to order these items.

Thus, SchoolStuff is the least expensive place to order 500 pens, 300 pencils, and 150 erasers.

Explanation:

Answer 2

The least expensive place to order 500 pens, 300 pencils, and 150 erasers is determined by matrix multiplication. The total cost is $33,500, with SchoolStuff providing the most economical option.

Let's set up the matrices to represent the costs and quantities of the items at each store.

Let:

-
`\( P \)` represent the matrix of prices,

-
`\( Q \)` represent the matrix of quantities,

-
`\( C \)` represent the matrix of costs.

The matrices are as follows:

`\[ P = \begin{bmatrix} 40 & 40 & 10 \\ \end{bmatrix} \]`

`\[ Q = \begin{bmatrix} 500 \\ 300 \\ 150 \\ \end{bmatrix} \]`

Now, to find the cost matrix
`\( C \)`, you can use matrix multiplication
`\( C = P * Q \).`

`\[ C = P * Q = \begin{bmatrix} 40 & 40 & 10 \\ \end{bmatrix} * \begin{bmatrix} 500 \\ 300 \\ 150 \\ \end{bmatrix} \]`

Performing the matrix multiplication:

`\[ C = \begin{bmatrix} 40 * 500 + 40 * 300 + 10 * 150 \end{bmatrix} \]`

`\[ C = \begin{bmatrix} 20000 + 12000 + 1500 \end{bmatrix} = \begin{bmatrix} 33500 \end{bmatrix} \]`

So, the least expensive place to order 500 pens, 300 pencils, and 150 erasers is where the cost is minimized, and in this case, it is a total cost of
`\( $33,500 \).`