How to set this up? "The RDA of zinc is 15mg; the RDA of ascorbic acid is 45mg..."

[allegansveritatem]

How to set this up? "The RDA of zinc is 15mg; the RDA of ascorbic acid is 45mg..."

​How to solve this with two systems of equations?:


The RDA of zinc is 15mg
The RDA of ascorbic acid is 45mg


Health bar A has 30 mg of ascorbic acid and 6 mg of zinc


Health bar B has 15mg of ascorbic acid and 3mg of zinc


How many oz of each must one eat to have exactly the RDA fore ascorbic acid and zinc?


I set this up thus:
x= oz of A Y= oz of B


30x + 15y = 45
6x + 3y = 15


This system results in a false statement thus:


30x + 15y = 45
-5 (6x) + -5 (3y) = -5 (15) which results in:


0 = -35.


Did I set this problem up wrong? Or is this saying that the problem is not solvable. Note that what is being asked is how much of a combination of the two bars ( stated thus: so much of bar A and so much of bar B) must one eat to get the exact requirements.

 

[stapel]

​How to solve this with two systems of equations?:

The RDA of zinc is 15mg
The RDA of ascorbic acid is 45mg

Health bar A has 30 mg of ascorbic acid and 6 mg of zinc [per ounce]
Health bar B has 15mg of ascorbic acid and 3mg of zinc [per ounce]

Note: The nutrition in Bar A is exactly twice the nutrition in Bar B.

​How many oz of each must one eat to have exactly the RDA for ascorbic acid and zinc?

I set this up thus:

. . .x: number of oz of A
. . .y: number of oz of B

. . .30x + 15y = 45
. . . .6x + 3y = 15

This system results in a false statement thus:

. . .30x + 15y = 45
. . .-5 (6x) + -5 (3y) = -5 (15)

which results in:

. . .0 = -35.

Did I set this problem up wrong? Or is this saying that the problem is not solvable. Note that what is being asked is how much of a combination of the two bars ( stated thus: so much of bar A and so much of bar B) must one eat to get the exact requirements.

Thank you for showing your work so nicely!

I agree with your set-up and your work. But let's back up to those two equations:

. . . . .30x + 15y = 45
. . . . . .6x + 3y = 15

Divide the first equation by 15 and the second equation by 3:

. . . . .2x + y = 3
. . . . .2x + y = 5

These are actually two parallel lines! They can't cross, so there can't be a solution. Maybe there's a typo in the exercise...? You might want to ask your instructor!

 

[allegansveritatem]

Note: The nutrition in Bar A is exactly twice the nutrition in Bar B.


Thank you for showing your work so nicely!

I agree with your set-up and your work. But let's back up to those two equations:

. . . . .30x + 15y = 45
. . . . . .6x + 3y = 15

Divide the first equation by 15 and the second equation by 3:

. . . . .2x + y = 3
. . . . .2x + y = 5



These are actually two parallel lines! They can't cross, so there can't be a solution. Maybe there's a typo in the exercise...? You might want to ask your instructor!

Thanks for reply. I see your point. But there has to be an answer to this problem, no? I mean they are asking how much of bar A and how much of bar B do you have to eat to get the RDA of ascorbic acid and zinc. So it is not a matter of lines...

Here is a copy of the problem in question:

 

[allegansveritatem]

Thanks for reply. I see your point. But there has to be an answer to this problem, no? I mean they are asking how much of bar A and how much of bar B do you have to eat to get the RDA of ascorbic acid and zinc. So it is not a matter of lines...

Here is a copy of the problem in question:View attachment 9404

Or can it be that the problem hinges on the word "exactly" ? There may be no way you can eat of those two bars in such a way to get exactly the RDA of each substance? You will of necessity get more of either ascorbic acid or zinc if you try. Could this result be so construed? Or have I missed something here?

 

[Dr.Peterson]

Or can it be that the problem hinges on the word "exactly" ? There may be no way you can eat of those two bars in such a way to get exactly the RDA of each substance? You will of necessity get more of either ascorbic acid or zinc if you try. Could this result be so construed? Or have I missed something here?

That's exactly right.

In fact, each bar has 5 times as much ascorbic acid as zinc, so no matter how much of each you eat, you will never get exactly 3 times as much ascorbic acid as zinc!

The book has a typo, or something equivalent.

 

[JeffM]

Thanks for reply. I see your point. But there has to be an answer to this problem, no? I mean they are asking how much of bar A and how much of bar B do you have to eat to get the RDA of ascorbic acid and zinc. So it is not a matter of lines...

Here is a copy of the problem in question:View attachment 9404

As stapel said, the problem as stated is insoluble.

ASSUME THERE IS A SOLUTION.

Therefore, to get exactly 45 mg of ascorbic acid, we calculate

\(\displaystyle 30a + 15b = 45 \implies 15b = 45 - 30a \implies b = 3 - 2a.\)

To get exactly 15 mg of zinc we calculate

\(\displaystyle 6a + 3b = 15 \implies 3b = 15 - 6a \implies b = 5 - 2a.\)

BUT THAT MEANS

\(\displaystyle 3 - 2a = 5 - 2a \implies 2a + 3 - 2a = 2a + 5 - 2a \implies 3 = 5.\)

But that conclusion is nonsense. So our initial assumption that there was a solution must be nonsense too.

The whole point of analytic geometry is to allow you to look at an exact visual analog of a numeric problem. If the lines that represent visually your equations never cross, there are no numbers that satisfy the equations.

EDIT: Yes it is "exactly" that makes the problem insoluble. So far I have seen you ask two questions, and both times your question involved a bad problem. Not your fault.

 

[allegansveritatem]

That's exactly right.

In fact, each bar has 5 times as much ascorbic acid as zinc, so no matter how much of each you eat, you will never get exactly 3 times as much ascorbic acid as zinc!

The book has a typo, or something equivalent.

I guess the author wanted to throw a curve here. He likes to mix it up.

 

[allegansveritatem]

As stapel said, the problem as stated is insoluble.

ASSUME THERE IS A SOLUTION.

Therefore, to get exactly 45 mg of ascorbic acid, we calculate

\(\displaystyle 30a + 15b = 45 \implies 15b = 45 - 30a \implies b = 3 - 2a.\)

To get exactly 15 mg of zinc we calculate

\(\displaystyle 6a + 3b = 15 \implies 3b = 15 - 6a \implies b = 5 - 2a.\)

BUT THAT MEANS

\(\displaystyle 3 - 2a = 5 - 2a \implies 2a + 3 - 2a = 2a + 5 - 2a \implies 3 = 5.\)

But that conclusion is nonsense. So our initial assumption that there was a solution must be nonsense too.

The whole point of analytic geometry is to allow you to look at an exact visual analog of a numeric problem. If the lines that represent visually your equations never cross, there are no numbers that satisfy the equations.

EDIT: Yes it is "exactly" that makes the problem insoluble. So far I have seen you ask two questions, and both times your question involved a bad problem. Not your fault.

I'm thinking the author, Robert Blixer, used the word "exactly" with the express purpose of showing what you are saying above. I have to say I like Blixer for the way he works hard to show the relevance of these techniques in "down home" situations.

 

[Dr.Peterson]

I'm thinking the author, Robert Blixer, used the word "exactly" with the express purpose of showing what you are saying above. I have to say I like Blixer for the way he works hard to show the relevance of these techniques in "down home" situations.

If you mean Robert Blitzer, I've taught from one of his books, and he does like applications and realism; but still, I would expect to see some flag warning you, however subtly, that "impossible" is a valid answer, and perhaps giving you a chance to learn from the experience rather than become frustrated. Have you stated the entire problem exactly as given? Are there instructions before a set of problems that might suggest some might be unsolvable? Is it marked as, say, a "critical thinking exercise"? Are there any exercises for which answers are given in the back of the book, which say they have no solution?

 

[JeffM]

I'm thinking the author, Robert Blixer, used the word "exactly" with the express purpose of showing what you are saying above. I have to say I like Blixer for the way he works hard to show the relevance of these techniques in "down home" situations.

I agree with Dr. Peterson.

It is a good thing to point out that it is possible to be given a problem that has no answer, to be given a problem that has multiple answers, to be given a problem that has infinite answers, and to be given problems that have no answer that can be expressed exactly in decimal form.

But if no clue is given that a problem in a text book does not have a small number of answers that can be expressed exactly, that problem is likely to demotivate many students. It is one thing psychologically for people who fully grasp that math has limits to run across an insoluble problem. Being assigned the task of solving the insoluble is very likely to have an adverse psychological effect on those already uncertain of their skills or the utility or comprehensibility of math.