help me with linear programing?

Question

Farm Stand sales you have 180 tomatoes and 15 onions left over from your garden. You want to use these to make jars of tomato sauce and jars of salas to sell at a farm stand. A jar of tomato sauce requires 10 tomatoes and 1 onion, and a jar of salsa requires 5 tomatoes and 1/4 onlion. You will make a profit of $2 on every jar of tomato sauce sold and a profit of $1.50 on every jar of salsa sold. The owner of the farm stand wants at least three times as many jars of tomato sauce as jars of salsa. How many jars of each should you make to maximize profit?

Answer 1

First, let's define the variables we're going to be solving for.
x= amount of tomato jars, y= amount of jars of salsa.

The objective function is always the easiest to figure out, because it's always about profit, so you're going to look for dollar signs.
>>"You will make a profit of $2 on every jar of tomato sauce sold and a profit of $1.50 on every jar of salsa sold."
which will translate into the equation C= 2x + 1.5y.

Constraints: Now, you always want to group together similar variables in a word problem. In this case:
A. tomatox + tomatoy must be less than or equal to the tomatoes you have
B. onionx + oniony must be less than or equal to the onions you have
which will translate into
A. 10x + 5y <= 180
B. 1x + 1/4 (or 0.25)y <= 15

Then, we'll translate the 3 times as many tomato jars as salsa jars. Remember the variables.
The tomato jars must be greater than or equal to three times the salsa jars.
x >= 3y

Now you graph. You find the vertexes, and plug them all into the objective function. Whichever number comes out the highest, you pick the coordinate points (x, y) that gave you that high number.

You can either graph manually, use a graphing calculator, or use this site:
http://people.hofstra.edu/Stefan_Waner/realworld/LPGrapher/lpg.html
However, don't depend on the site too much, because you'll have to change around the equations so the script can process it and help you solve it. This may alter the answer at times.
For ex., you have to change the x >= 3y to 3y - x <= 0.
You'll get 13.846154, and 4.615385. You would round it to 14, 5, when the correct answer is 14, 4.

So 14 jars of tomato sauce, and 4 jars of salsa sauce for the maximum profit of $34. I hope that wasn't too long and confusing. Even though you may have turned in this homework quite a while ago, this should help you with future linear systems/linear programming problems.

Answer 2

Since you make more profit per jar of sauce (50 cents more per jar), you should maximize the sauce. Also one onion covers 10 tomatoes for sauce, but 20 for salsa.

Assuming you have to make at least one jar of salsa, make 4 jars of salsa so you're using an entire onion and 20 tomatoes. That leaves 14 onions for sauce using 140 tomatoes for 14 jars of sauce.

14 jars of sauce @ $2 profit each is $28 profit
4 jars of salsa @ 1.50 profit each is $6 profit
Total $34 profit and 20 tomatoes leftover to enjoy or find more onions for!

That's the only combination using whole onions that meets the criteria of at least 3 times as many jars of sauce as salsa.

Answer 3

Umm...do you have the recipes?