math is not fun

Factual Questions
1

So, I have been working on this last problem for three hours. Can you say obsessive compulsive personality. I sure can.

Selling plain and caramel popcorn balls. 300 hours to prepare the balls for sale. It takes 1/10 of an hour to prepare the plain popcorn balls and 4/5 of an hour for the caramel ones. 250 hours to package. Plain takes 1/5 of an hour, caramel 9/20 of an hour.
$260 to spend on supplies. Plain costs 10 cents/lb, caramel 40 cents/lb. Maximize profit if plain balls are 35 cents, caramel ones 75 cents.

I ended up with three equations:

x<=3000-8y
x<=1,250-45y
x<=1,600-4y

Then I graphed. The last equation does not intersect the other two. The first two intersect at (330,335) which is a solution for the first equation, but not the other two. I tweaked the numbers around and still cannot come up with something that satisfies all equations, and I have no clue what to do with the line that refuses to help form a polygon. I know there are other ways to solve this, I just don’t know what they are. Could anyone PLEASE give me a hand here?

2

Do you mean 45/20 y in your second equation and 2600 in the third?

Remember, to satisfy these equations you don’t have to be on the line, merely below/left of it. Draw the three lines on the graph. Shade the right/top of each line to indicate infeasibility. Shade -ve x and y. The origin and the (positive) points on the axes closest to the origin where a line intersects the axis should be some corners of the unshaded polygon, which should be bounded at the top and right by two or three of the lines.

It’s entirely possible that one restraint is not needed, eg. try x<=1,y<=2,y<=3. Every point satisfied by the first two equations is also satisfied by the last. Conversely there will be no extreme satisfying points on the third line because there are all ruled out by y<=1.

Good luck

3

You’ve got

Maximise:

0.25 x + 0.35 y

Subject to:

0.1 x + 0.8 y <= 300
0.2 x + 0.45 y <= 250
0.1 x + 0.4 y <= 260

Or, stated the way you have, you want to graph the lines:

x = 3000 - 8 y
x = 1,250 - 2.25 y
x = 2,600 - 4 y

4

sigh

I am getting tired. Actually the original problem should have read $160 to spend. And yes, it should have been 45/20y in the second equation.

5

There you go.

x = 800, y = 200.

6

Crap.

x = 200
y = 350.

7

Hmmm, I tried both answers out and each only fulfills two of the three constraints. Shouldn’t the correct answer fulfill all three of the equations?

8

No. They’re inequality constraints.

Draw the lines. Shade the portion of the plabe above and to the right of each line.

You are left with four possible solutions.

One at the x axis, one at the y axis, and one at each of the two intersection points I identified above.

9

plabe = plane, obviously.

10

Ok, thanks alot. I really used to understand this stuff. Anyway, your first answer was the right one, x=800 y=200. The income is maximized for .35x + .75y. Now I am off to mathforum.org to try to understand the whole thing better. I do appreciate your help.

11

You should buy a 200 pocket scientific PDA. Then you could be confused scientifically.

12

Actually, NOW it seems that correct answer is:

x=1,250
y=0

No caramel popcorn at all maximizes profits. Only have 165 to spend.

13

I keep giving out misinformatiion. I only have 160 to spend, not 165. Sorry. Perhaps if I was not so sloppy I would figure these things out myself.