A pharmacist mixes together three types of tablets. Each tablet A contains,among other things, 13 mg of niacin and 11 IU of vitamin E. The amounts for a tablet Bare 18 mg and 14 IU, and for a tablet C are 23 mg and 36 IU. Determinehow many of each tablet there are if there are 225 total tablets, 4300 mg of niacin, and5200 IU of vitamin E.
Linear programming: A pharmacist mixes together three types of tablets...
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What are your thoughts? What have you tried? How far did you get? Where are you stuck? For instance, you started by picking variables for the unknowns. What variables did you pick? What were their definitions? And so forth.
Please be complete. Thank you!
What I have so far..
So I created a table and this is what I have so far:
These are the equations I came up with
A+B+C=225 (since there are a total of 225 tablets)
13A+18B+23C=4300 (equation for niacin) ..............edited
11A+14B+36C=5200 (equation for vitamin E)
I am stuck from here, is this the right approach/correct equations? I am getting confused on how to solve the equations.
So I created a table and this is what I have so far:
| NIACIN | VITAMIN E | |
| Tablet A | 13 | 11 |
| Tablet B | 18 | 14 |
| Tablet C | 23 | 36 |
| Total | 4300 | 5200 |
These are the equations I came up with
A+B+C=225 (since there are a total of 225 tablets)
13A+18B+23C=4300 (equation for niacin) ..............edited
11A+14B+36C=5200 (equation for vitamin E)
I am stuck from here, is this the right approach/correct equations? I am getting confused on how to solve the equations.
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What you have so far is perfectly fine. If I were solving this problem, my next step would be to start manipulating the equations. You know that a + b + c = 225. And you know that 13a + 18b + 23c = a + a + a + ... (13 a's) + b + b + ... (18 b's) + c + c + ... (23 c's). And that, of course, still equals 4300. So, what happens if you rearrange the new form of the second equation, such that it's (a + b + c) + (a + b + c) + ...? Then make the appropriate substitution for each instance of a + b + c. Try continuing on from there, and see where that gets you.
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What have been taught to solve this type problem?
Have you been taught Gauss elimination? Kramer's method?
Are you allowed to use software like MS-excel?
Yes, it is the correct approach. You now have 3 equations in three unknowns. You can write the augmented matrix as
\(\displaystyle \begin{pmatrix}
1& 1& 1& 225\\
13& 18& 23& 4300\\
11& 14& 36& 5200\\
\end{pmatrix}
\)
Now reduce that to row echelon form to find the solution. The first thing you can do is subtract 13 times row 1 from row 2 to get
\(\displaystyle \begin{pmatrix}
1& 1& 1& 225\\
0& 5& 10& 1325\\
11& 14& 36& 5200\\
\end{pmatrix}
\)
Now divide row 2 by 5 to get
\(\displaystyle \begin{pmatrix}
1& 1& 1& 225\\
0& 1& 2& 275\\
11& 14& 36& 5200\\
\end{pmatrix}
\)
and continue from there.