Answer :
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]
### 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]
