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 19 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?
$
Now this is a linear programming problem (not the same chapter as the other ones you asked)
make a of A and b of B
constraints
3a+4b </= 1000 pounds iron
6a+3b </= 19*60 or 1140 minutes labor
graph those (a on x and b on y)
line 1 (iron)
(333,0) and (0,250) solution on or below)
line 2 (labor)
(190,0) and (0,380) (solution on or below)
intersection at (104,172)
so test 3 points
(0,250)
(104,172)
(190,0)
with profit = 2 x+1.5 y
point 1
p = 2(0) + 1.5*250 = $375
point 2
p = 2(104)+1.5(172) = $ 466 winner
point 3
p = 2(190) +1.5(0) = $380
point 1 p =
thanks sooooo much!
To maximize Kane's profit, we can set up a linear programming problem.
Let's define the variables:
x = number of model A hibachis produced
y = number of model B hibachis produced
Now let's set up the constraints based on the resources available:
3x + 4y ≤ 1000 (constraint for cast iron)
6x + 3y ≤ 19 (constraint for labor-hours)
Since we can't produce a negative number of hibachis, the variables are also bound to be non-negative:
x ≥ 0, y ≥ 0
Next, let's set up the objective function, which represents the profit:
Profit = 2x + 1.5y
To find the maximum profit, we need to solve this linear programming problem.
One approach is to graph the feasible region formed by the constraints and find the corner points, then evaluate the objective function at each corner point. The corner point that gives the maximum profit is the optimal solution.
However, since there are only a few constraints in this problem, we can use a simpler method called the Corner Point Method or the Vertex Method.
We will solve the system of equations formed by each pair of constraints and evaluate the objective function at each solution to determine the maximum profit.
Let's solve the system of equations for each pair of constraints:
1. Solve 3x + 4y = 1000 and 6x + 3y = 19
By solving this system, we get x = 200 and y = 150/7 (approximately 21.43).
2. Solve 3x + 4y = 1000 and y = 0
By solving this system, we get x = 333.33 and y = 0.
3. Solve 4y = 1000 and 6x + 3y = 19
By solving this system, we get x = 0 and y = 250/3 (approximately 83.33).
Now let's evaluate the objective function at each solution:
1. Profit = 2 * 200 + 1.5 * (150/7) ≈ $743.52
2. Profit = 2 * 333.33 + 1.5 * 0 = $666.66
3. Profit = 2 * 0 + 1.5 * (250/3) ≈ $416.66
Therefore, to maximize Kane's profit, the division should produce approximately:
model A hibachis: 200 hibachis
model B hibachis: 150/7 (approximately 21.43) hibachis
The largest profit the company can realize is approximately $743.52.
To solve this problem, we need to use linear programming.
Let's define the variables:
Let A be the number of model A hibachis produced.
Let B be the number of model B hibachis produced.
We want to maximize the profit, which is given by the equation:
Profit = 2A + 1.5B
Now, let's consider the constraints:
1. The available cast iron is limited to 1000 lb.
We know that producing each model A hibachi requires 3 lb of cast iron, and each model B hibachi requires 4 lb. Therefore, the cast iron constraint can be written as:
3A + 4B ≤ 1000
2. The available labor hours are limited to 19.
Producing each model A hibachi requires 6 minutes (which is 0.1 hours) of labor, and each model B hibachi requires 3 minutes (which is 0.05 hours) of labor. Therefore, the labor constraint can be written as:
0.1A + 0.05B ≤ 19
3. We cannot produce a negative number of hibachis.
A, B ≥ 0
Now that we have defined the objective function and constraints, we can use a linear programming solver to find the optimal solution.
By solving this linear programming problem, the solver will find the values of A and B that maximize the profit, subject to the given constraints.
Upon solving, the optimal solution will provide the number of hibachis to be produced for each model (A and B) and the corresponding maximum profit.
To find the solution, you can use a linear programming solver tool, such as Excel's Solver or an online LP solver.
I hope this explanation helps you understand how to approach this problem using linear programming.