The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day with the following stocks? (Use matrix inverses to find the solutions.)
(a)    1050 quarts of pineapple juice, 840 quarts of orange juice, 430 quarts of kiwi juice
PineOrange	     
150
 
Correct: Your answer is correct.
 gal
PineKiwi	     
250
 
Correct: Your answer is correct.
 gal
OrangeKiwi	     
180
 
Correct: Your answer is correct.
 gal


(b)    770 quarts of pineapple juice, 920 quarts of orange juice, 430 quarts of kiwi juice
PineOrange	     
100
 
Correct: Your answer is correct.
 gal
PineKiwi	     
190
 
Correct: Your answer is correct.
 gal
OrangeKiwi	    1060 
Incorrect: Your answer is incorrect.
 gal


(c)    A quarts of pineapple juice, B quarts of orange juice, C quarts of kiwi juice
PineOrange	     
  
Incorrect: Your answer is incorrect.
 gal
PineKiwi	     
  
Incorrect: Your answer is incorrect.
 gal
OrangeKiwi	     
  
Incorrect: Your answer is incorrect.
 gal



We are tasked with finding how many gallons of each juice blend Arctic Juice Company can make, using matrix inverses. We are given three juice blends: PineOrange, PineKiwi, and OrangeKiwi. The juice compositions are:

1. **PineOrange:** 2 quarts pineapple, 2 quarts orange (per gallon)
2. **PineKiwi:** 3 quarts pineapple, 1 quart kiwi (per gallon)
3. **OrangeKiwi:** 3 quarts orange, 1 quart kiwi (per gallon)

### Step 1: Define Variables

Let:
- \( x_1 \) = gallons of PineOrange
- \( x_2 \) = gallons of PineKiwi
- \( x_3 \) = gallons of OrangeKiwi

We need to formulate this problem using a system of equations. The total amount of juice (pineapple, orange, kiwi) available will be expressed in terms of \( x_1 \), \( x_2 \), and \( x_3 \).

### Step 2: Set up System of Equations

Based on the juice compositions, we can write the following system:

- **Pineapple Juice:**
  \[
  2x_1 + 3x_2 = \text{{total pineapple juice available}}
  \]
- **Orange Juice:**
  \[
  2x_1 + 3x_3 = \text{{total orange juice available}}
  \]
- **Kiwi Juice:**
  \[
  x_2 + x_3 = \text{{total kiwi juice available}}
  \]

### Step 3: Matrix Representation

We can express this system as a matrix equation \( A \mathbf{x} = \mathbf{b} \), where:

\[
A = \begin{pmatrix} 2 & 3 & 0 \\ 2 & 0 & 3 \\ 0 & 1 & 1 \end{pmatrix}, \quad
\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}, \quad
\mathbf{b} = \begin{pmatrix} \text{total pineapple} \\ \text{total orange} \\ \text{total kiwi} \end{pmatrix}
\]

### Part (a): 1050 quarts pineapple, 840 quarts orange, 430 quarts kiwi

For part (a), the system is:

\[
\begin{pmatrix} 2 & 3 & 0 \\ 2 & 0 & 3 \\ 0 & 1 & 1 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = \begin{pmatrix} 1050 \\ 840 \\ 430 \end{pmatrix}
\]

We solve this system using matrix inverses to find \( x_1 \), \( x_2 \), and \( x_3 \).

Let me calculate the matrix inverse and solve for the values.For part (a), the company can produce:

- 150 gallons of **PineOrange**,
- 250 gallons of **PineKiwi**,
- 180 gallons of **OrangeKiwi**.

Now, let's proceed with part (b) where the stocks are 770 quarts of pineapple, 920 quarts of orange, and 430 quarts of kiwi. I will solve the system for these values.It seems like I can’t do more advanced data analysis right now. Please try again later. If you have any other questions or need help, feel free to ask!

Here are some follow-up questions to explore:

1. What happens if we increase the available stock of pineapple juice?
2. How does the ratio of pineapple to kiwi juice impact the blend proportions?
3. What mathematical properties make the matrix inverse method suitable here?
4. How would the problem change if the ratios in each blend were different?
5. Can we solve this system using methods other than matrix inversion?

**Tip:** Matrix inverses are useful for solving systems of linear equations when the coefficient matrix is square and invertible.

The Arctic Juice Company makes three juice blends: PineOrange, using 2 quarts of pineapple juice and 2 quarts of orange juice per gallon; PineKiwi, using 3 quarts of pineapple juice and 1 quart of kiwi juice per gallon; and OrangeKiwi, using 3 quarts of orange juice and 1 quart of kiwi juice per gallon. The amount of each kind of juice the company has on hand varies from day to day. How many gallons of each blend can it make on a day with the following stocks? (Use matrix inverses to find the solutions.) (a) 1050 quarts of pineapple juice, 840 quarts of orange juice, 430 quarts of kiwi juice. (b) 770 quarts of pineapple juice, 920 quarts of orange juice, 430 quarts of kiwi juice. (c) A quarts of pineapple juice, B quarts of orange juice, C quarts of kiwi juice.

Explore how to use matrix inverses to solve a system of equations involving juice blends. Learn how to calculate the gallons of PineOrange, PineKiwi, and OrangeKiwi juice blends using stocks of pineapple, orange, and kiwi juice. This example problem is designed for students in Grades 10-12 and covers concepts in linear algebra and matrix inversion.

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