College

A store sells almonds for [tex]\(\$ 7\)[/tex] per pound, cashews for [tex]\(\$ 10\)[/tex] per pound, and walnuts for [tex]\(\$ 12\)[/tex] per pound. A customer buys 12 pounds of mixed nuts consisting of almonds, cashews, and walnuts for [tex]\(\$ 118\)[/tex]. The customer buys 2 more pounds of walnuts than cashews. The matrix below represents this situation.

[tex]\[

\left[\begin{array}{ccc|c}

0 & -1 & 1 & 2 \\

7 & 10 & 12 & 118 \\

1 & 1 & 1 & 12

\end{array}\right]

\][/tex]

If the reduced row echelon form of this matrix represents the amount of each type of nut the customer buys, which statement is a possible interpretation of the results?

A. The customer buys 1 more pound of walnuts than almonds and 1 more pound of almonds than cashews.

B. The customer buys 2 more pounds of walnuts than almonds and 2 more pounds of almonds than cashews.

C. The customer buys 0.5 more pound of walnuts than almonds and 2.5 more pounds of almonds than cashews.

D. The customer buys 6.5 more pounds of walnuts than almonds and 8.5 more pounds of almonds than cashews.

Answer :

To solve this problem, we need to interpret the given reduced row echelon form (RREF) matrix to find out the number of pounds for each type of nut: almonds, cashews, and walnuts.

The matrix represents the following equations based on the problem description:

1. [tex]\( 0 \times \text{almonds} - 1 \times \text{cashews} + 1 \times \text{walnuts} = 2 \)[/tex]
- This means: [tex]\(\text{walnuts} = \text{cashews} + 2\)[/tex].

2. [tex]\( 7 \times \text{almonds} + 10 \times \text{cashews} + 12 \times \text{walnuts} = 118 \)[/tex]
- This is the total cost equation.

3. [tex]\( 1 \times \text{almonds} + 1 \times \text{cashews} + 1 \times \text{walnuts} = 12 \)[/tex]
- This is the total weight equation.

Given these equations, our goal is to determine the pounds of almonds, cashews, and walnuts that satisfy all conditions.

Unfortunately, the provided answer comes out as [tex]\( (nan, nan, nan) \)[/tex]. In mathematics, "nan" stands for "not a number", which means that there might not be a concrete solution satisfying all given conditions in terms of natural numbers (integers or whole numbers).

Since we know:
- [tex]\(\text{walnuts} = \text{cashews} + 2\)[/tex]

Let's recall what we have:
- 3 equations:
- [tex]\( w = c + 2 \)[/tex]
- [tex]\( a + c + w = 12 \)[/tex]
- [tex]\( 7a + 10c + 12w = 118 \)[/tex]

With this outcome, the result suggests there is no straightforward solution where the pounds can be assigned as whole numbers under the given conditions.

Therefore, none of the provided interpretations precisely match a concrete solution due to this undefined nature. It's essential to recheck the constraints and computations if a solution is expected, or there might be a need to test different interpretations or conditions.