Answered

The number of items produced by a company partly varies as the volume of water ([tex]v[/tex]) and partly varies jointly as the number of machines ([tex]M[/tex]) and the quantity of raw materials ([tex]Q[/tex]).

With 4 machines, 6 silos of raw materials, and 15 cubic meters of water, the company produced 129 boxes of items in a day.

With 4 machines, 6 silos of raw materials, and 19 cubic meters of water, the company produced 148 boxes of items.

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Note: The text has been reformatted and corrected for readability, but the question may still need additional context or clarification regarding the mathematical relationship being described.



Answer :

GinnyAnswer
Sure, let’s break down the problem step by step.

### Step 1: Define the relationship

We are told that the number of items produced by a company partly varies directly as the volume of water ([tex]\(v\)[/tex]) and partly varies jointly as the number of machines ([tex]\(M\)[/tex]) and the quantity of raw materials ([tex]\(Q\)[/tex]).

We can express this relationship with the equation:

[tex]\[ \text{produced} = k_1 \cdot v + k_2 \cdot M \cdot Q \][/tex]

### Step 2: Formulate the equations from given data

We are given the quantities in two different scenarios and the resulting number of items produced.

Scenario 1:
- Volume of water ([tex]\(v_1\)[/tex]) = 15 m³
- Number of machines ([tex]\(M\)[/tex]) = 4
- Quantity of raw materials ([tex]\(Q\)[/tex]) = 6
- Items produced ([tex]\(produced_1\)[/tex]) = 129

Using these, we can write the first equation:
[tex]\[ 129 = k_1 \cdot 15 + k_2 \cdot 4 \cdot 6 \][/tex]
[tex]\[ 129 = 15 \cdot k_1 + 24 \cdot k_2 \][/tex]

Scenario 2:
- Volume of water ([tex]\(v_2\)[/tex]) = 19 m³
- Number of machines ([tex]\(M\)[/tex]) = 4
- Quantity of raw materials ([tex]\(Q\)[/tex]) = 6
- Items produced ([tex]\(produced_2\)[/tex]) = 148

Using these, we can write the second equation:
[tex]\[ 148 = k_1 \cdot 19 + k_2 \cdot 4 \cdot 6 \][/tex]
[tex]\[ 148 = 19 \cdot k_1 + 24 \cdot k_2 \][/tex]

### Step 3: Solve the system of linear equations

We now have two linear equations:

1. [tex]\( 129 = 15 \cdot k_1 + 24 \cdot k_2 \)[/tex]
2. [tex]\( 148 = 19 \cdot k_1 + 24 \cdot k_2 \)[/tex]

We need to solve these simultaneously to find the values of [tex]\(k_1\)[/tex] and [tex]\(k_2\)[/tex].

### Step 4: Subtract the first equation from the second

Subtracting the first equation from the second removes [tex]\(k_2\)[/tex] term, allowing us to solve for [tex]\(k_1\)[/tex] first.

[tex]\[ 148 - 129 = (19 \cdot k_1 + 24 \cdot k_2) - (15 \cdot k_1 + 24 \cdot k_2) \][/tex]
[tex]\[ 19 = 4 \cdot k_1 \][/tex]
[tex]\[ k_1 = \frac{19}{4} \][/tex]

Now substitute [tex]\(k_1 = \frac{19}{4}\)[/tex] back into the first equation to find [tex]\(k_2\)[/tex]:

[tex]\[ 129 = 15 \cdot \frac{19}{4} + 24 \cdot k_2 \][/tex]
[tex]\[ 129 = \frac{285}{4} + 24 \cdot k_2 \][/tex]
[tex]\[ 129 = 71.25 + 24 \cdot k_2 \][/tex]
Subtract 71.25 from both sides,

[tex]\[ 57.75 = 24 \cdot k_2 \][/tex]
[tex]\[ k_2 = \frac{57.75}{24} \][/tex]
[tex]\[ k_2 = \frac{77}{32} \][/tex]

### Step 5: Conclusion

The constants [tex]\(k_1\)[/tex] and [tex]\(k_2\)[/tex] for the given production model are:

[tex]\[ k_1 = \frac{19}{4} \][/tex]
[tex]\[ k_2 = \frac{77}{32} \][/tex]

Therefore, the number of items produced by the company can be modeled with the equation:

[tex]\[ \text{produced} = \frac{19}{4} \cdot v + \frac{77}{32} \cdot M \cdot Q \][/tex]