2. A factory manufactures laser printers and inkjet printers. The factory cannot manufacture more than 26 printers per month. Due to the demand for inkjet printers, the company must manufacture at least 10 inkjet printers per month. A laser printer requires 70 hours to manufacture and an inkjet printer requires 35 hours. A maximum of 1 050 hours is available for manufacturing per month. The manufactuf makes a profit of R2 700 on each laser printer and R900 on each inkjet printer sold. a) Write down the constraints in the form of inequalities. b) Draw the graphs on graph paper and indicate the feasible region. c) How many of each type of printer must be manufactured per mont d) Determine the maximum profit.

Question

UpStudy AI · Accepted Answer

To solve this problem, we will follow the steps outlined in the prompt.

### Step 1: Extract Known Conditions

1. **Total printers manufactured per month**: Cannot exceed 26.
2. **Inkjet printers manufactured per month**: Must be at least 10.
3. **Manufacturing hours**:
   - Laser printer: 70 hours each.
   - Inkjet printer: 35 hours each.
   - Total available hours: 1,050 hours.
4. **Profit**:
   - Laser printer: R2,700 each.
   - Inkjet printer: R900 each.

### Step 2: Write Down the Constraints in the Form of Inequalities

Let:
- $$ x $$ = number of laser printers manufactured
- $$ y $$ = number of inkjet printers manufactured

The constraints can be formulated as follows:

1. Total printers constraint:
   $$
   x + y \leq 26
   $$

2. Minimum inkjet printers constraint:
   $$
   y \geq 10
   $$

3. Manufacturing hours constraint:
   $$
   70x + 35y \leq 1050
   $$

4. Non-negativity constraints:
   $$
   x \geq 0, \quad y \geq 0
   $$

### Step 3: Draw the Graphs and Indicate the Feasible Region

To find the feasible region, we will graph the inequalities. However, since I cannot draw graphs directly, I will describe how to do it:

1. **Graph the line $$ x + y = 26 $$**:
   - Intercepts: $$ (26, 0) $$ and $$ (0, 26) $$.

2. **Graph the line $$ y = 10 $$**:
   - This is a horizontal line at $$ y = 10 $$.

3. **Graph the line $$ 70x + 35y = 1050 $$**:
   - Rearranging gives $$ y = -2x + 30 $$.
   - Intercepts: $$ (15, 0) $$ and $$ (0, 30) $$.

4. **Identify the feasible region**:
   - The feasible region is bounded by the lines and must satisfy all inequalities.

### Step 4: Determine the Number of Each Type of Printer to Maximize Profit

To maximize profit, we need to evaluate the profit function:
$$
P = 2700x + 900y
$$

We will find the vertices of the feasible region and evaluate the profit at each vertex.

1. **Vertices of the feasible region**:
   - Intersection of $$ x + y = 26 $$ and $$ y = 10 $$:
     $$
     x + 10 = 26 \implies x = 16 \quad \Rightarrow (16, 10)
     $$
   - Intersection of $$ x + y = 26 $$ and $$ 70x + 35y = 1050 $$:
     $$
     y = 26 - x \implies 70x + 35(26 - x) = 1050
     $$
     $$
     70x + 910 - 35x = 1050 \implies 35x = 140 \implies x = 4 \quad \Rightarrow (4, 22)
     $$
   - Intersection of $$ y = 10 $$ and $$ 70x + 35y = 1050 $$:
     $$
     70x + 35(10) = 1050 \implies 70x + 350 = 1050 \implies 70x = 700 \implies x = 10 \quad \Rightarrow (10, 10)
     $$

2. **Evaluate profit at each vertex**:
   - At $$ (16, 10) $$:
     $$
     P = 2700(16) + 900(10) = 43200 + 9000 = 52200
     $$
   - At $$ (4, 22) $$:
     $$
     P = 2700(4) + 900(22) = 10800 + 19800 = 30600
     $$
   - At $$ (10, 10) $$:
     $$
     P = 2700(10) + 900(10) = 27000 + 9000 = 36000
     $$

### Step 5: Determine the Maximum Profit

The maximum profit occurs at the vertex $$ (16, 10) $$ with a profit of R52,200.

### Summary of Results

- **a)** Constraints:
  $$
  x + y \leq 26, \quad y \geq 10, \quad 70x + 35y \leq 1050, \quad x \geq 0, \quad y \geq 0
  $$
- **b)** The feasible region is determined by the intersection of the inequalities.
- **c)** The factory should manufacture 16 laser printers and 10 inkjet printers per month.
- **d)** The maximum profit is R52,200.
The factory should manufacture 16 laser printers and 10 inkjet printers each month to achieve a maximum profit of R52,200.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions