Solve using information given.

Question

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 21 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?
$____ 

Is there any raw material left over? (If so, give the amount remaining. If not, enter 0.)

cast iron _____lb

labor _____ min

Answer 1

Just like your other question, this is a type of problem called "linear programming." If you follow the same steps we followed for that problem, you should wind up with the correct answer. There are a couple differences. First, we won't be working through two problems at once (Machine I and Machine II), since model production is bounded by both material and labor together. And second, since there are four bounds on production, your solution set will be a quadrilateral (not a triangle), so there will be four points of intersection. Here are the steps, as a reminder:



STEP 1: Define variables



STEP 2: Write inequalities that bound production

These will include x ≥ 0, y ≥ 0, an inequality that bounds the number of each model produced by minutes of labor, and an inequality that bounds the number of each model produced by pounds of material.

 

STEP 3: Graph each inequality and find the region of overlap (solution set)

The solution set for this problem will probably be a quadrilateral bounded by the two axes and your two other equations.



STEP 4: Find points of intersection

You'll likely be finding four points of intersection. Check your graph to figure out which line intersections you're interested in. The intersection of x = 0 and y = 0 will be one vertex, and another will likely be where the other two equations intersect.

STEP 5: Write an equation for profit and test each vertex for maximization

 

 

 

Let me know if you get stuck!