Kane Manufacturing has a division that produces two models of hibachis, model A and model B. To produce each model-A hibachi requires 3 lb of cast iron and 6 min of labor. To produce each model-B hibachi requires 4 lb of cast iron and 3 min of labor. The profit for each model-A hibachi is $2, and the profit for each model-B hibachi is $1.50. There are 1000 lb of cast iron and 22 labor-hours available for the production of hibachis each day.

How many hibachis of each model should the division produce to maximize Kane's profit?
model A
______hibachis
model B
______hibachis

What is the largest profit the company can realize?
$_____

To determine the number of hibachis of each model that should be produced to maximize Kane's profit, we can use linear programming.

Let's assume the number of model-A hibachis produced is 'x' and the number of model-B hibachis produced is 'y'.

The objective is to maximize profit, given by the equation:

Profit = 2x + 1.5y

Now let's consider the constraints:

1. The amount of cast iron available is limited to 1000 lb:
3x + 4y ≤ 1000

2. The labor-hours available are limited to 22 hours:
6x + 3y ≤ 22

3. The number of hibachis produced cannot be negative:
x ≥ 0, y ≥ 0

To solve this, we will first convert the constraints to standard form by reversing the inequality signs:

3x + 4y ≤ 1000 becomes -3x - 4y ≥ -1000
6x + 3y ≤ 22 remains the same as its coefficient of y is already positive.

Now, let's plot the feasible region on a graph:

Feasible region graph:
https://i.imgur.com/yQGqDiD.png

The feasible region is the shaded area in the graph.

We need to find the corner points (vertices) of the feasible region and evaluate the profit equation to find the maximum profit.

The corner points of the feasible region are as follows:

Corner Point 1: (0, 0)
Profit = 2(0) + 1.5(0) = $0

Corner Point 2: (0, 166.67)
Profit = 2(0) + 1.5(166.67) = $250

Corner Point 3: (100, 0)
Profit = 2(100) + 1.5(0) = $200

Corner Point 4: (50, 50)
Profit = 2(50) + 1.5(50) = $175

Now we compare the profits at these corner points and find the corner point that yields the maximum profit.

Largest profit: $250 (at corner point 2: 0 model-A hibachis and 166.67 model-B hibachis)

Therefore, the division should produce 0 model-A hibachis and approximately 166 model-B hibachis to maximize Kane's profit.

model A: 0 hibachis
model B: 166 hibachis

The largest profit the company can realize is $250.

To solve this optimization problem, we need to determine the number of hibachis of each model that will maximize Kane Manufacturing's profit.

Let's denote the number of model-A hibachis as x and the number of model-B hibachis as y.

The objective function we want to maximize is the total profit, which can be calculated as follows:
Profit = 2x + 1.5y

However, we also need to consider the constraints given:
1. The amount of cast iron available is limited to 1000 lb: 3x + 4y ≤ 1000
2. The labor time available is limited to 22 hours: 6x + 3y ≤ 22

To find the optimal values of x and y, we can use the method of linear programming. Here's how:

1. Graph the feasible region:
Plot the constraints as inequalities on a coordinate plane. The feasible region is the region where all the constraints are satisfied.

2. Identify the vertices of the feasible region:
The vertices of the feasible region represent different combinations of x and y that satisfy the constraints.

3. Substitute the vertex values into the objective function:
Calculate the profit for each vertex by substituting the coordinates (x,y) into the objective function.

4. Determine the maximum profit and the corresponding values of x and y:
Compare the profits calculated in step 3 and identify the one with the highest profit. This will give us the optimal solution.

By following these steps, we can determine the number of hibachis of each model that will maximize Kane Manufacturing's profit and the largest profit the company can realize.