Word problem: systems of inequalities and equations

[Crookshanks]

I'm stuck on question 2 of this problem:

A cook is puzzling over the number of pounds of food he should purchase in order to minimize his cost. He has always bought his food from a small health food store in town. The store sells two types of mixtures. Both of these mixtures contain the three ingredients needed, but the cook needs his own special ratio of these ingredients to meet the requirements of a certain diet. The chart below shows that the diet is to be made up of at least 9 grams of Vitamin B-12, at least 30 grams of calcium, and at least 24 grams of iron. As the chart also shows the ratios for each of the three ingredients in the two mixtures the store sells is not the same as what the cook needs. The cook wants to minimize his costs, and have the minimum requirements met once he combines the two mixtures and cooks up the entire blend.

Dietary Constraints
Vitamin B-12 Calcium Iron
Min. Reqs 9 grams 30 grams 24 grams
Mixture A
= $3.00/lb 1 g per pound 6 g per pound 8 g per pound
Mixture B
= $5.00/lb 3 g per pound 5 g per pound 3 g per pound


Question 1: First set up an equation for the total cost the cook will have to pay for the two mixtures he will buy. This will be called the cost function. Let x = the number of pounds of Mixture A, and y = the number of pounds of Mixture B.

My answer: F(c) = 3x + 5y


Question 2:Next, develop three inequalities utilizing the cook's dietary constraints.

THIS IS WHERE IM STUCK
I came up with
3x + 1y > 9
6x + 5y > 30
8x + 3y > 24
but these don't seem right. I think the (3g per pound, 5g per pound, etc.. is confusing it for me) What am I missing here? Thanks.

 

[Gene]

You got x & y mixed up in the first equation..
You get x grams of b-12 from x pounds of A
You get 3y grams of b-12 from y pounds of B
so x+3y>9
Checking: 9 lbs of A or 3 lbs of B will do it. x=9 ot y=3 will do it. Okay.
I see that you did the other two correctly.
Now graph the three equations with A along the bottom and B along the side, excluding the areas below the lines. (They were all >s) The answers are the intersection points. There are four points. (Don't forget the intercepts.) Try the x & y values in your cost function to find the cheapest.

 

[Crookshanks]

Would these be > or >= (greater than or equals to)? Do these make much of a difference when graphing?

 

[tkhunny]

Read "at least" to mean ">=", but it makes little difference in either the solution or the graph.

 

[Crookshanks]

Going further into this question.

Draw a graph for the solution of the three inequalities (this does not need to be turned in). Shading will indicate the solution to each inequality. In this case, since the wording of the inequalities was "at least" 9 grams of Vitamin B-12, etc., the inequalities will utilize the = symbol. The shaded area that satisfies all three inequalities will form a solution area bounded by the segments of straight lines.
Next, calculate the coordinates of each point in the solution area where any two of the boundary lines intersect, as well as where any of the boundary lines cross the axes.

a. 1st point of intersection is:
b. 2nd point of intersection is:
c. x-intercept is:
d. y-intercept is.

Explain how to obtain these.

Now I found the x intercept to b 9,0 and the y intercept to be 0,8. Just eyeballing (the graph I drew) it it appears that the 1st point of intersection is 1.5 , 5.5 and the second is 3.5 , 2 . It doesnt explain in my book how to find the intersection points though. Is this done by eyeballing or is there an actual way to determine this?

 

[Gene]

Each of the three equations will have an x & y intercept. After you shade it, two of each will be in the unshaded area and no longer count. THE y intercept will be the shaded one. Same for THE x.
Better check your graph. The x intercept is good. Is the y intercept a typo?
You could find the exact intercepts by solving the three equations in pairs to get three intercepts, one in the unshaded area to be ignored. Or check your eyeballs by substituting into the two that meet there.
Your intersections are pretty rough. One trick is to draw another graph of only the square (or four squares) the crossing is in (say ten by ten makeing each square on this one 0.1 wide) to get another decimal.