Question

a chemist combines two substance to make a new drug. The number of doses of the drug is given by 4√x*√y, where x is the amount of the first substance used, and y is the amount of the second (in grams). The first substance cost twice as much as the second on (per gram)
Use Lagrange Multipliers to find the exact amounts of each substance that should be used for producing 100 doses of the drug at a minimal cost

Answer 1

I don't remember learning Lagrange Multipliers, but it's been a while...I may have just foprgotten them. I'm using this site to figure it out....Give me a minute to type it in though.

http://en.wikipedia.org/wiki/Lagrange_multiplier

Answer 2

\[(x,y)=((25\sqrt{2})/2,25\sqrt{2})\]Because you want the number of doses to be 100, and the function that gives the numberof doses is given by\[d(x,y)=4\sqrt{xy}\]Let's set this equal to 100 to get our constraint.
\[4\sqrt{xy}=100\]\[\sqrt{xy}=25\]\[xy=625\]

Now, the price per gram of the substances is irrelevant, but we want to minimize the cost. So setting the cost of the cheaper one to 1, the more expensive one would be 2.

Effectively, we want to minimize the funciton f(x,y)=y+2x given the constraint xy=625.

So Lagrange funciton is

\[\Lambda(x,y,\lambda)=y+2x+\lambda(xy-625)\]
Now we need to find the system of partial derivaives of this over all three variables. and set them equal to 0
\[\delta \Lambda/\delta x=2+\lambda y =0\]\[\delta \Lambda/\delta y=1+\lambda y =0\]\[\delta \Lambda/\delta \lambda=xy-625 =0\]
from the first two we get\[y=-2/\lambda\]and \[x=-1/\lambda\]plugging these into the third and solving for lambda, we get\[\lambda=\pm(\sqrt{2}/25)\].

Plugging these back into the equations for x and y, we see that\[x=\pm(25\sqrt{2})/2\]\[y=\pm(50\sqrt{2})/2=\pm25\sqrt{2}\]
Since only positive amounts of the drugs work, we eliminate the negative solutions giving us\[(x,y)=((25\sqrt{2})/2,25\sqrt{2})\]

Answer 3

:) I havent even begun to play with LaGrange yet, its on my to do list for next year tho ;)

Answer 4

lol...then you're one up on me. It didn't look familiar, but seemed like the sort of thing I would have done at some point. thanks though :)