Linear programming: find profit-maximizing num. of paintings

[Ti-Pro.doc.]

Hassan is an artist who specializes in geometric designs. He is trying to get ready for the street fair next month. Hassen paints both watercolors and pastels. Each type of picture takes him about the same time to paint. He figures he has time to make a total of at most 16 pictures. The material for each pastel will cost him $5 dollars and the material for each watercolor will cost him $15 dollars. He has $180 to spend on materials. Hassen makes a profit of $40 on each pastel and a profit of $100 on each watercolor.

Let "x" be the number of watercolors and "y" be the number of pastels.

How would I graph and solve this inequality? :?:

 

[stapel]

This isn't one "inequality" (singular). This is a set of inequalities.

The first step is to take the given relationships and the given variables, and translate them into inequalities. Have you studied word-problem translations yet?

Once you have the set of inequalities, you have to graph them. Have you studied that yet?

Once you have the various lines graphed, you have to find the coordinates of the corners of the polygon (the "region") that the inequalities have marked off. Have you studied how to solve systems of equations yet?

Once you have found the coordinates of the corners of the region, you plug them into the optimization formula that you would have found in the first, "translating", step above.

(As you can see, linear programming is an involved and multi-step process. You'll need to let us know on how many of these steps you are needing lessons.)

Thank you.

Eliz.

 

[Ti-Pro.doc.]

i meant plural. i just need the equations because nothing i tryed is working

 

[stapel]

...nothing i tryed is working

Please reply showing all your work and reasoning, so that we may try to find your error (if any). You should have the usual minimum inequalities, a "time" inequality, and a "materials cost" inequality, along with a "profit" optimation equation.

Please be specific. Thank you.

Eliz.

 

[Ti-Pro.doc.]

ok so far we know a few obvious equations: x ≥ 0, y ≥ 0, x + y ≤ 16

here is an equation i wasn't quite sure on:

180 ≥ 15x + 5y

Here is the profit equation: P = 40y + 100x

 

[Ti-Pro.doc.]

 

[stapel]

ok so far we know a few obvious equations: x ≥ 0, y ≥ 0, x + y ≤ 16

Did your instructor and/or text never mention that these are actually "inequalities"? "Equations" have "equals" signs in them; "inequalities" are the ones with the inequality ("less than" or "greater than") symbols.

Your first two inequalities would appear to be the usual basic constraints: You cannot paint a negative number of canvasses.

Your third inequality would appear to the the "time" constraint: You have time for painting no more than sixteen canvasses.

This leaves, as mentioned earlier, the "materials cost" inequality, which would appear to be your fourth inequality:

180 ≥ 15x + 5y

(The reason it was suggested that you show your reasoning was that then you would understand how you arrived at your inequalities. As it is, I have no idea how you created these inequalities, when you don't appear to know where they came from...?)

Here is the profit equation: P = 40y + 100x

Yes; this is the optimization equation.

Eliz.

 

[Ti-Pro.doc.]

I meant "inequalities". I'm just so used to graphing equations that I accidentally reverted to the wrong term.

Thank you for your help.

 

[stapel]

I meant "inequalities".

Excellent! (Often enough, students do not know the difference, so it seemed wise to ask, rather than risk causing unnecessary confusion.)

Glad I could help!

Eliz.