anonymous
  • anonymous
please help!! Solve the optimization problem. Maximize P = xyz with x + y = 12 and y + z = 12, and x, y, and z ≥ 0. p = ??
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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asnaseer
  • asnaseer
express P in terms of x only then find the value of x that makes dP/dx=0 use this value to find the max value for P
anonymous
  • anonymous
x = 12-y and z= 12-y
anonymous
  • anonymous
(12-y)^2 (y)

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asnaseer
  • asnaseer
yes - you can do it that way as well
asnaseer
  • asnaseer
now you need to find the value for y that makes dP/dy=0
anonymous
  • anonymous
im stuck at that part
asnaseer
  • asnaseer
do you know how to use the product rule when differentiating?
asnaseer
  • asnaseer
if not - this might help: http://en.wikipedia.org/wiki/Product_rule
asnaseer
  • asnaseer
or else expand the expression out and then differentiate
anonymous
  • anonymous
can you help me with the first step
asnaseer
  • asnaseer
what method are you planning to use?
asnaseer
  • asnaseer
product rule or expand and then differentiate?
anonymous
  • anonymous
the easiest way to do it
asnaseer
  • asnaseer
do you know how to differentiate an expression like \(3x^2+4x^3+6x\)?
anonymous
  • anonymous
6(1+x+2x^2)
asnaseer
  • asnaseer
ok - so first expand your expression:\[(12-y)^2 (y)\]which is usually written in this form:\[y(12-y)^2\]
anonymous
  • anonymous
right i got that part, then what, find the derivative?
asnaseer
  • asnaseer
yes - but can you please first expand that expression - what do you get?
asnaseer
  • asnaseer
I am /assuming/ you do not know how to use the product or chain rule. which is why I first asked if you at least knew how to differentiate a general polynomial expression. since you do, then for you, the easiest would be to first expand that expression to get a polynomial expression and then differentiate it.
anonymous
  • anonymous
y(12-y)(12-y) = y(144-24y+y^2)
asnaseer
  • asnaseer
correct so far - now distribute y into the braces
anonymous
  • anonymous
144y-24y^2+y^3
asnaseer
  • asnaseer
perfect - now differentiate with respect to y
asnaseer
  • asnaseer
again, here I am /assuming/ that you are aware that to find the max/min of a polynomial expression (if one exists) if to differentiate it and set the resulting expression to zero. that will give you the value for the unknown for which the original expression has a max/min (or point of inflection)
asnaseer
  • asnaseer
*is to differentiate it and...
asnaseer
  • asnaseer
so you have now found that:\[P=144y-24y^2+y^3\] you now need to calculate dP/dy
anonymous
  • anonymous
3(y^2-16y+48) .... is that correct?
asnaseer
  • asnaseer
correct, now find the value(s) of y that will make this expression equal to zero
asnaseer
  • asnaseer
you should be able to factor the expression inside the braces
anonymous
  • anonymous
y= 4 , y= 12
asnaseer
  • asnaseer
correct - now one of those y values will give the maximum value for P and the other will probably represent a minimum value for P
asnaseer
  • asnaseer
do you know how to tell whether you have found a max/min/point of inflection using the second derivative test?
anonymous
  • anonymous
no
asnaseer
  • asnaseer
what have you been taught about local maximums and minimums? I just want to get a idea of what approach to show you that best matches what you have been taught.
anonymous
  • anonymous
not much, i actually have an assignment due very soon so i was just hoping what the next step would be after finding y=12, y=4
asnaseer
  • asnaseer
well substituting y=12 into your expression (\(P=144y-24y^2+y^3\)) will give you one value for P. And substituting y=4 into this expression will give you another value for P. One of these will represent the maximum value that P can have and the other will represent the minimum value that P can have (given the original constraints in the problem).
anonymous
  • anonymous
ok then how would i find Z?
asnaseer
  • asnaseer
when you started the problem you established that: x = 12-y and z= 12-y so just substitute the two value for y into these expressions to get the corresponding two values each for x and z.
asnaseer
  • asnaseer
but first you need to work out which of the two y values gives you the maximum P value. Then use just that vaue of y to get x and z.
anonymous
  • anonymous
so i got 256 and 3456 as my answers
asnaseer
  • asnaseer
what do those values represent? how did you calculate them?
anonymous
  • anonymous
y (4) = 256, y(12) = 3456, plugged both into 144y-24y^2+ y^3
asnaseer
  • asnaseer
your calculation is not correct for y=12
asnaseer
  • asnaseer
use the factored form you originally got:\[P=y(12-y)^2\]this should make it easier to calculate the values for P.
anonymous
  • anonymous
wouldnt that equal to 0?
asnaseer
  • asnaseer
for y=12, yes
asnaseer
  • asnaseer
so you have now found: y=12 gives P=0 y=4 gives P=256
asnaseer
  • asnaseer
so which one do you think is the maximum?
anonymous
  • anonymous
256
asnaseer
  • asnaseer
correct - so problem solved. :)
anonymous
  • anonymous
ok, can i ask you another question, a lot easier than this.. please?
asnaseer
  • asnaseer
It is very late where I am and I need to catch some sleep so that I can get to work tomorrow fully awake. I suggest you post your new question in the list on the left - there are plenty of very good people on this site who will surely come to your aid.
anonymous
  • anonymous
ok thanks for the help
asnaseer
  • asnaseer
yw :)

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