Question

The snack shop makes 3 mixes of nuts in the following proportions.

mix I: 6 lbs peanuts, 2 lbs cashews, 2 lbs pecans.

Mix II: 5 lbs peanuts, 3 lbs cashews, 2 lbs pecans.

Mix III: 3 lbs peanuts, 4 lbs cashews, 3 lbs pecans.

they received an order for 25 of mix I, 18 of mix II, and 35 of mix III. write the matrices A & B for which AB gives the total number of lbs of each nut required to fill the order.

Answer 1

`A=\begin{bmatrix}6 & 5 & 3\\ 2 & 3 & 4\\ 2 & 2 & 3\end{bmatrix}`

`B=\begin{bmatrix}25\\ 18\\ 35\end{bmatrix}`

Explanation:

It is given that the snack shop makes 3 mixes of nuts in the following proportions.

Mix I: 6 lbs peanuts, 2 lbs cashews, 2 lbs pecans.

Mix II: 5 lbs peanuts, 3 lbs cashews, 2 lbs pecans.

Mix III: 3 lbs peanuts, 4 lbs cashews, 3 lbs pecans.

they received an order for 25 of mix I, 18 of mix II, and 35 of mix III.

We need to find the matrices A & B for which AB gives the total number of lbs of each nut required to fill the order.

Mix I Mix II Mix III

peanuts 6 5 3

cashews 2 3 4

pecans 2 2 2

`A=\begin{bmatrix}6 & 5 & 3\\ 2 & 3 & 4\\ 2 & 2 & 3\end{bmatrix}`

`B=\begin{bmatrix}25\\ 18\\ 35\end{bmatrix}`

The product of both matrices is

`AB=\begin{bmatrix}6 & 5 & 3\\ 2 & 3 & 4\\ 2 & 2 & 3\end{bmatrix}\begin{bmatrix}25\\ 18\\ 35\end{bmatrix}`

`AB=\begin{bmatrix}6\cdot \:25+5\cdot \:18+3\cdot \:35\\ 2\cdot \:25+3\cdot \:18+4\cdot \:35\\ 2\cdot \:25+2\cdot \:18+3\cdot \:35\end{bmatrix}`

`AB=\begin{bmatrix}345\\ 244\\ 191\end{bmatrix}`

Therefore matrix AB gives the total number of lbs of each nut required to fill the order.