Question

1. buddy’s nut shop sells two varieties of mixed nuts. the mixes contain different amounts of peanuts (p), hazelnuts (h), cashews (c), and walnuts (w). matrix ( p ) below gives the portion of the total weight of each mix for each type of nut. the owner plans to change the mixes soon, and so decides to combine what is left of the two mixes and sell this combined mixture. matrix ( n ) gives the current amounts for each mix.

\begin{array}{c} \text{regular} \\ \text{deluxe mix} end{array} \begin{bmatrix} 45% & 20% & 15% & 20% \\ 20% & 30% & 25% & 25% end{bmatrix} = p
\text{weight in lbs} \begin{bmatrix} 16 & 4 end{bmatrix} = n
( \boldsymbol{a.} ) what is the meaning of matrix ( np )? find this matrix.
( \boldsymbol{b.} ) what is the amount of hazelnuts in the combined mixture?

Explanation:

Part (a)

Step1: Understand Matrix Dimensions and Multiplication

Matrix \( N \) is a \( 1 \times 2 \) matrix (\( [16\quad4] \)) and matrix \( P \) is a \( 2 \times 4 \) matrix. When multiplying matrices \( N \) and \( P \) (in the order \( NP \)), the resulting matrix will be \( 1 \times 4 \). The element in the \( i \)-th row and \( j \)-th column of \( NP \) is the dot product of the \( i \)-th row of \( N \) and the \( j \)-th column of \( P \). This will give the total weight (in lbs) of each type of nut (peanuts, hazelnuts, cashews, walnuts) in the combined mixture.

Step2: Perform Matrix Multiplication

First, convert the percentages to decimals:
\( P=

$$\begin{bmatrix}0.45 & 0.20 & 0.15 & 0.20\\0.20 & 0.30 & 0.25 & 0.25\end{bmatrix}$$

\)
\( N =

$$\begin{bmatrix}16 & 4\end{bmatrix}$$

\)

Calculate the first element (peanuts):
\( 16\times0.45 + 4\times0.20 = 7.2 + 0.8 = 8.0 \)

Second element (hazelnuts):
\( 16\times0.20 + 4\times0.30 = 3.2 + 1.2 = 4.4 \)

Third element (cashews):
\( 16\times0.15 + 4\times0.25 = 2.4 + 1.0 = 3.4 \)

Fourth element (walnuts):
\( 16\times0.20 + 4\times0.25 = 3.2 + 1.0 = 4.2 \)

So, \( NP=

$$\begin{bmatrix}8.0 & 4.4 & 3.4 & 4.2\end{bmatrix}$$

\) (or in percentage terms, but since we multiplied by weights in lbs and percentages (as decimals), the result is in lbs. The meaning of \( NP \) is the total weight (in pounds) of each type of nut (peanuts, hazelnuts, cashews, walnuts) in the combined mixture of the regular and deluxe mixes.

Step1: Identify the Hazelnut Column in \( NP \)

From the matrix \( NP \) calculated in part (a), the second element (since hazelnuts are the second column in matrix \( P \)) represents the total weight of hazelnuts in the combined mixture.

Step2: Extract the Value

From \( NP=

$$\begin{bmatrix}8.0 & 4.4 & 3.4 & 4.2\end{bmatrix}$$

\), the weight of hazelnuts is \( 4.4 \) pounds.

Answer:

(for part a):
The matrix \( NP \) represents the total weight (in pounds) of each type of nut (peanuts, hazelnuts, cashews, walnuts) in the combined mixture. The matrix \( NP=

$$\begin{bmatrix}8.0 & 4.4 & 3.4 & 4.2\end{bmatrix}$$

\) (or with appropriate units, the weights of peanuts, hazelnuts, cashews, walnuts in lbs respectively).

Part (b)