Hi below is the question I am having so much troube with. I have been
told the answer is D but I have no idea how to get it. If anyone knows
how to it will be much appreciated also please explain all working!
Especially what the notation min {su, po + 2so} means (I think its an
easy question I just dont know what that notation means!!!)


Ben runs a cookie factory. His cookies are made with sugar, peanut
oil, and soybean oil. The
number of boxes of cookies that he produces is f (su, po, so) = min
{su, po + 2so}, where
su is the number of bags of sugar, po the number of canisters of
peanut oil, and so the number
of canisters of soybean oil that he uses. The price of a bag of sugar
is $12. The price of a
canister of peanut oil is $6. The price of a canister of soybean oil
is $19. If Ben makes 254
boxes of cookies in the cheapest way possible, how many canisters of
soybean oil will he
use?
A. 84.67
B. 127
C. 169.33
D. 0
E. 42.33

Hi  irish_traveller!!


Well, this is a tricky question and the answer results very short and
may be a surprising one.

The answer is D as you was told, that is if Ben makes 254 boxes of
cookies in the cheapest way possible, he will use zero canisters of
soybean oil.

Why? you ask. It is easy, first note that sugar and oils are not the
only ingredients of the cookies, there are others; but the amount of
production depends only on 3 variables the number of bags of sugar,
the number of canisters of peanut oil, and so the number of canisters
of soybean oil used.
Well, when Ben wants to produce cookies in the cheapest way possible
he only use the cheapest ingredients.
It is easy to see that peanut oil and soybean oil are substitutes,
that is you can use one instead the other or both together; but prices
are different, each canister of soybean oil cost $19 and the price of
a canister of peanut oil is $6.
Ben want to make cookies in the cheapest way possible so he will use
only the needed amount of cheapest oil, in this case peanut oil, and
no soybean due its high cost. This is why the answer is zero canister
of soybean will be used.


Regarding the meaning of "f(su,po,so) = min {su, po + 2so}":
The statement of the problem says that the number of boxes of cookies
produced is function of the following variables or factors:
su: number of bags of sugar used,
po: number of canisters of peanut oil used, and
so: number of canisters of soybean oil used.

The production function is called f and the notation "f(su,po,so)"
indicates which are the independent variables of the function.
The notation "f(su,po,so) = min {su, po + 2so}" (better written as
"min{su,(po+2*so)}", where the asterisk denotes the common product,
like in 2*3=6) describes the function f indicating which values the
production reach according to the values of the independent variables.
"min" means minimum, then f(su,po,so) = min{su,(po+2*so)} means that:
The number of number of boxes of cookies that produced is equal to THE
MINIMUM VALUE BETWEEN THE FOLLOWING 2 NUMBERS:
a) su
b) po+2*so

For example if he uses 10 bags of sugar, 5 canisters of peanuts oil
and 3 canisters of soybean oil, we have that:
a) su = 10
b) po+2*so = 5 + 2*3 = 11 ;
then:
f(su,po,so) = min{su,(po+2*so)} =
            = min{10,11} =
            = 10
Therefore Ben will produce 10 boxes of cookies.


I hope this helps you to understand the problem and its solution. feel
free to request for a clarification if you need it.


Regards,
livioflores-ga

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Request for Answer Clarification by irish_traveller-ga on 28 May 2006 06:57 PDT

Thank you so much for your answer, you've really helped me out! :D
One thing I need clarifcation for is "it is easy to see peanut oil and
soybean oil are perfect substitutes", how would a student know this
from the question?

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Request for Answer Clarification by irish_traveller-ga on 28 May 2006 07:22 PDT

I have posted another question on this topic and would really
appreciate your help on that too

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Clarification of Answer by livioflores-ga on 28 May 2006 08:13 PDT

Hi again!!

You asked:
"One thing I need clarifcation for is "it is easy to see peanut oil and
soybean oil are perfect substitutes", how would a student know this
from the question?"

There is no way to know that from the statement, this is why it is a
tricky question. May be I did not use the more appropriate expression
at this point, but both are oils and not an essential part of the
recipe (if you know a little about cooking).

Hope this helps.

Regards,
livioflores-ga

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Request for Answer Clarification by irish_traveller-ga on 28 May 2006 08:39 PDT

Thanks livioflores, if ur on now could you take a look at my other question?

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Clarification of Answer by livioflores-ga on 28 May 2006 19:10 PDT

Thank you for the good rating!!

Very helpful, good long answer