Help advanced math problem involving inequalities and graphing .

[tngirl]

Two workers (Choot and Wanlah) assemble hand grenades and landmines for a notoriously frugal warlord.
The adventurers learn that the warlord demands a minimum of 12 hand grenades and 30 land mines every day, and has figured out how to get what he wants while paying his two workers the least possible (while still honouring their agreed-upon hourly wages). While Lara and Harrison can't tell how many hours each worker spends assembling bombs, or how many bombs they actually make each day, they did learn something about the workers' wages and how efficiently they both work. They wrote down this information in the following note:
*
Choot- 7/hr - 3 grenades plus 3 landmines per hour

Wanlah- 6/hr- 1 grenade plus 4 landlines per hour
*
1. How long do the two bomb makers work each day?
2. How many grenades and land mines do they make each day?
*

 

[tngirl]

Yes it was worded by my teacher. I am in grade 12. My teacher said it is a logical reasoning problem involving linear inequalities, feasible region and constraints.

 

[JeffM]

Yes it was worded by my teacher. I am in grade 12. My teacher said it is a logical reasoning problem involving linear inequalities, feasible region and constraints.

More formally it is a linear programming problem.

Do you know anything about constraining equations, objective function, feasible region, optimum vertex?

 

[tngirl]

Yes, I am familiar with constraining equations and feasible region. So far, I have come up with these two equations for the first question. 3x + 3y ≤ 12 and 1x + 4y ≤ 30. I am considering finding the P.O.I of these two equations. Am I on the right path?

 

[JeffM]

Yes, I am familiar with constraining equations and feasible region. So far, I have come up with these two equations for the first question. 3x + 3y ≤ 12 and 1x + 4y ≤ 30. I am considering finding the P.O.I of these two equations. Am I on the right path?

Sort of in the general vicinity of the path.

First, in every word problem, whether it is algebra, calculus, whatever, NAME your variables in writing. This reduces the burden on your memory and let's you communicate with others. It may also help give you partial credit on a test.

So x = hours worked by Choot

y = hours worked by Wanlah

c = cost incurred by warlord. You did not identify this variable.

Second step in every word problem, translate EVERY condition of the problem into mathematical form. This may be hard because some conditions may not be explicit. Some may be implied or require general knowledge.

What variable is to be optimized?

In linear programming, there are almost always implicit non-negativity constraints. In this problem there are two implied non-negativity constraints plus two additional implied constraints.

\(\displaystyle 0 \le x.\) Hours worked daily cannot be negative.

\(\displaystyle x \le 24.\) Hours worked daily cannot exceed 24.

\(\displaystyle 0 \le y.\)

\(\displaystyle y \le 24.\)

Now you made a mistake on the other two constraints. The warlord wants AT LEAST 12 hand grenades so

\(\displaystyle 3x + y \ge 12.\) You had less than or equals.

The warlord wants AT LEAST 30 landmines so

\(\displaystyle 3x + 4y \ge 30.\) You had less than or equals.

Now there is something else very important in a programming problem. Do you know what that is called?

I want to reiterate that this process for solving word problems works for word problems in almost any field of math.

 

[tngirl]

Sort of in the general vicinity of the path.

First, in every word problem, whether it is algebra, calculus, whatever, NAME your variables in writing. This reduces the burden on your memory and let's you communicate with others. It may also help give you partial credit on a test.

So x = hours worked by Choot

y = hours worked by Wanlah

c = cost incurred by warlord. You did not identify this variable.

Second step in every word problem, translate EVERY condition of the problem into mathematical form. This may be hard because some conditions may not be explicit. Some may be implied or require general knowledge.

What variable is to be optimized?

In linear programming, there are almost always implicit non-negativity constraints. In this problem there are two implied non-negativity constraints plus two additional implied constraints.

\(\displaystyle 0 \le x.\) Hours worked daily cannot be negative.

\(\displaystyle x \le 24.\) Hours worked daily cannot exceed 24.

\(\displaystyle 0 \le y.\)

\(\displaystyle y \le 24.\)

Now you made a mistake on the other two constraints. The warlord wants AT LEAST 12 hand grenades so

\(\displaystyle 3x + y \ge 12.\) You had less than or equals.

The warlord wants AT LEAST 30 landmines so

\(\displaystyle 3x + 4y \ge 30.\) You had less than or equals.

Now there is something else very important in a programming problem. Do you know what that is called?

I want to reiterate that this process for solving word problems works for word problems in almost any field of math.

Okay. I understand the first part, but I don't understand what you mean by programming problem? Perhaps, our teacher used a different term.

 

[tngirl]

That wasn't my point; I was simply "skook up" with stuff like:
"hand grenades and landmines for a notoriously frugal warlord".

Why not something more peaceful Anyhow, forget it...

Hahaha. I see what you mean.

 

[JeffM]

Okay. I understand the first part, but I don't understand what you mean by programming problem? Perhaps, our teacher used a different term.

This problem is an example of what is called a linear programming problem because the objective function to be optimized and the constraining functions are all linear functions. The solution to a linear programming problems always occurs at the vertex of two or more constraining functions. To determine which vertex optimizes the objective function is basically what linear programming is all about. When there are a small number of vertices, you can make that determination by hand. So what is the objective function? Where are the vertices of the constraining functions?

 

[tngirl]

This problem is an example of what is called a linear programming problem because the objective function to be optimized and the constraining functions are all linear functions. The solution to a linear programming problems always occurs at the vertex of two or more constraining functions. To determine which vertex optimizes the objective function is basically what linear programming is all about. When there are a small number of vertices, you can make that determination by hand. So what is the objective function? Where are the vertices of the constraining functions?

I am just thinking outloud here but, wouldnt there be 5 or 6 points of intersection of the feasible region. Then, I would find the x and the y of each intersection, and then test each point (x,y) into the cost equation which could be, c = 7x + 6y . The cost to the boss will be the smallest value of c that you get from those (x,y) points. And the x and the y will be the point, (x,y) that gave you the smallest value of c. I am so confused on this right now.

 

[JeffM]

I am just thinking outloud here but, wouldnt there be 5 or 6 points of intersection of the feasible region. Then, I would find the x and the y of each intersection, and then test each point (x,y) into the cost equation which could be, c = 7x + 6y . The cost to the boss will be the smallest value of c that you get from those (x,y) points. And the x and the y will be the point, (x,y) that gave you the smallest value of c. I am so confused on this right now.

I haven't worked out the feasible region so I am not sure exactly how many points of intersection there are, but, yes, there could well be five or six such points. (In a real practical problem, there could be millions of them.) Your cost function is an example of what is called the objective function in the lingo of linear programming; it is what you want to optimize, in this case to minimize. Your objective is to minimize cost. And yes you have properly defined the cost function.

The mathematical fact is that, in a linear programming problem, the objective function will be minimized at one of the intersection points of the constraining functions that define the feasible region. So yes, you just test the cost function at those points. In a practical problem with millions of intersections, there are ways to find the optimum without looking at all the points.

Edit: There may not be a feasible solution. If there is one, then the solution occurs at an intersection.