Completely stumped

[abcd]

The problem: A small store been asked to make cabinet doors in two sizes. Each day the shop can make no more than 6 large doors or no more than 12 small doors. Each day the shop receives 48 feet of wood. It takes 6 feet of wood to make a large door and 4 feet of wood to make a small door.The store makes a profit of $12 per large door and $20 per small door. Use the method of linear programming to determine how many of each door they should produce daily to maximize their profits from the doors.

What I've done so far:

A. Declare a set of variables for the store's situation. Write a system of inequalities that models the situation.

Large doors require 6 feet of wood per door, or 6x.
Small doors require 4 feet, or 4y.
6x+4y <= 48

Large doors provide a profit of $12 each, or 12x.
Small doors provide a profit of $20 each, or 20y.
12x+20y

What I'm having trouble with:

B. Describe the solution graph by giving the boundaries of the solution area as inequalities and state the relationship of the solution to the inequality, ie: above, below, to the left, to the right, etc.

This is the only step that I'm really having trouble with; I don't understand the question at all. Please help.

 

[tutor_joel]

Part A is not finished. You need to come up with all possible outcomes. ie.

\(\displaystyle x \le 8\)

\(\displaystyle y \le 12\)

12x + 20y doesn't tell you anything since it's not set up as an inequality. Once you get all of your inequalities, simply graph them all and shade the appropriate sections and see what section is shaded the most. (or not shaded if you do it the opposite way). Does this make sense?

 

[mmm4444bot]

A. Declare a set of variables for the store's situation. You did not declare your variables x and y.

They want to see explicit definitions for each symbol x and y.

Write a system of inequalities that models the situation. There are three inequalities to write.

6x + 4y <= 48 This is one of them.

12x + 20y This is not an inequality. This is an expression that represents the profit made from selling x big doors and y small doors.

For the remaining two inequalities, think about x and y. Can x be any Real number? Can y be any Real number?

B. Describe the solution graph by giving the boundaries of the solution area as inequalities

It seems to me that this is a duplicated request for the same system of equations requested in part A. I'm not sure why they've asked for it twice. Perhaps, I'm misreading something.

and state the relationship of the solution to the inequality, ie: above, below, to the left, to the right, etc.

Well, you first need to find the solution, before you can start comparing it's location on the graph to the various lines.

Show us the system, and your solution. We'll go from there.

 

[abcd2]

12x + 20y doesn't tell you anything since it's not set up as an inequality. Once you get all of your inequalities, simply graph them all and shade the appropriate sections and see what section is shaded the most. (or not shaded if you do it the opposite way). Does this make sense?

So would it be 12x+20y <= total profit? :/

I'm still pretty confused...


So, x is a large door, y is a small one.

I think they can be any real number... maybe

x <= 48/6
y <= 48/4

Is that right?

 

[mmm4444bot]

So would it be 12x+20y <= total profit

Yes. We can let the symbol P represent profit.

P = 12x + 20y

The exercise asks you to find the (x, y) pair that makes P as big as possible (maximum profit) under the given contraints on x and y.

So, x is a large door, y is a small one.

No, x is not a door!

The symbols x and y represent numbers:

x = the number of large doors produced

y = the number of small doors produced

These two equations are the "declarations" for the variables. They tell us the meaning of the symbols x and y.

I think they can be any real number... maybe Really? Can we produce negative numbers of doors?

x <= 48/6

y <= 48/4 This can't be right. x and y must each be Whole numbers. (Who would purchase a fraction of a door?)

We're given the upper limits on x and y:

"Each day the shop can make no more than 6 large doors or no more than 12 small doors."

So, x must be 6 or smaller, and y must be 12 or smaller, and neither x nor y can be less than zero.

Does this make sense?

You can write these inequalities in the following form (you fill in the four numbers):

? ? x ? ?

? ? y ? ?

You previously wrote a good inequality to model the constraint on the amount of wood that's available.

6x + 4y ? 48

Now, you've got some lines to graph.

You don't need to graph the lines x = 0 and y = 0 (the lower limits on x and y) because the axes serve to keep the graph in Quadrant I.

So, you need to graph the following three lines.

x = 6

y = 12

6x + 4y = 48

Shade the appropriate sides of each, to find the solution region.

Evaluate P for the (x, y) pairs at the vertices of this region, to test for the pair that gives the largest value for P.

 

[garf]

abcd:
Just remember, once you have graphed everything, the intersection coordinates of the boundaries (lines), one of those ponits will be the one that maximizes the profit.