SOLUTION: A company produces three products, each of which must be processed through different departments. The table, in the next page, summarizes the hours required per unit of each produc

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Question 1160887: A company produces three products, each of which must be processed through different departments. The table, in the next page, summarizes the hours required per unit of each product in each department. The weekly capacities are stated for each department in terms of work hours available. Determine whether there is a combination of the three products which could be produced monthly so as to consume full hours available per week.
Department. ProductA. Product B. Product C. Hours available per week
A. 2. 3.5. 3. 1200
B. 3. 2.5. 2. 1150
C. 4. 3. 2. 1400




Answer by CPhill(2285) About Me  (Show Source):
You can put this solution on YOUR website!
To determine if there is a production combination of Products A, B, and C that consumes all available weekly hours in each department, we can set up and solve a system of linear equations.
Let:
* $x$ = units of Product A produced per week
* $y$ = units of Product B produced per week
* $z$ = units of Product C produced per week
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### Step 1: System of Equations
Using the hours required per unit in each department:
$$\begin{aligned} \text{Department A:} \quad & 2x + 3.5y + 3z = 1200 \\ \text{Department B:} \quad & 3x + 2.5y + 2z = 1150 \\ \text{Department C:} \quad & 4x + 3y + 2z = 1400 \end{aligned}$$
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### Step 2: Solve the System
1. **Subtract Department B's equation from Department C's equation to eliminate $z$:**
$$(4x + 3y + 2z) - (3x + 2.5y + 2z) = 1400 - 1150$$

$$x + 0.5y = 250 \implies x = 250 - 0.5y$$

2. **Eliminate $z$ between Department A and Department B:**
Multiply Department B's equation by $1.5$:
$$1.5(3x + 2.5y + 2z) = 1.5(1150) \implies 4.5x + 3.75y + 3z = 1725$$

Now, subtract Department A's equation from this result:
$$(4.5x + 3.75y + 3z) - (2x + 3.5y + 3z) = 1725 - 1200$$

$$2.5x + 0.25y = 525$$

3. **Substitute $x = 250 - 0.5y$ into the simplified equation:**
$$2.5(250 - 0.5y) + 0.25y = 525$$

$$625 - 1.25y + 0.25y = 525$$

$$625 - y = 525 \implies \mathbf{y = 100}$$

4. **Solve for $x$:**
$$x = 250 - 0.5(100) \implies \mathbf{x = 200}$$

5. **Solve for $z$ using Department B's equation:**
$$3(200) + 2.5(100) + 2z = 1150$$

$$600 + 250 + 2z = 1150$$

$$850 + 2z = 1150$$

$$2z = 300 \implies \mathbf{z = 150}$$


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### Conclusion
**Yes**, there is a unique combination of the three products that fully utilizes all available weekly hours in all three departments:
* **Product A:** **200 units** per week
* **Product B:** **100 units** per week
* **Product C:** **150 units** per week