Math Is Fun Forum

tongzilla

Member

Registered: 2006-06-19

Posts: 18

How to solve this problem?

Problem:

1) Both

and

is always greater than zero. Are there any restictions needed on

for this to be true?

===========================================
EDIT (21 OCT 07):

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:

1) What restrictions (i.e. the general rule) are needed on

and

so that there are some range of values of

and

to make both

and

positive?

2) What is the restriction (i.e. the general rule) for

and

such that no matter what combination of

and

is chosen (but still sums to 100),

and

is always negative?

3) How can you prove your answers to questions 1) and 2) ?

Last edited by tongzilla (2007-10-20 21:05:32)

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How to solve this problem?

Similarly

tongzilla

Member

Registered: 2006-06-19

Posts: 18

Re: How to solve this problem?

Similarly

Jane, not sure if your solution works.

First, lets say

and

(I made these values up, but they satisfy my initial condition that they must be greater than one)

then this implies

and

So this doesn't satisfy my condition that

Second, given the these borderline values of

and

(72.46 and 32.26),

and

are both negative (plug it into my initial equations to see yourself), and therefore can not be greater than zero. So the problem hasn't been solved.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How to solve this problem?

But your chosen values of x[sub]A[/sub] and x[sub]B[/sub] will not satisfy the further conditions

Will they? Just because

doesn’t mean it’s up to you to choose any values greater than 1 for them if further conditions are being specified. You need to ensure that they satisfy those further conditions as well.

In this case, if you want P[sub]A[/sub] and P[sub]B[/sub] to be positive, certain values of x[sub]A[/sub] and x[sub]B[/sub] will not work.

Last edited by JaneFairfax (2007-10-20 10:15:19)

tongzilla

Member

Registered: 2006-06-19

Posts: 18

Re: How to solve this problem?

But your chosen values of x[sub]A[/sub] and x[sub]B[/sub] will not satisfy the further conditions

Will they? Just because

doesn’t mean it’s up to you to choose any values greater than 1 for them if further conditions are being specified. You need to ensure that they satisfy those further conditions as well.

In this case, if you want P[sub]A[/sub] and P[sub]B[/sub] to be positive, certain values of x[sub]A[/sub] and x[sub]B[/sub] will not work.

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:

1) What restrictions (i.e. the general rule) are needed on

and

so that there are some range of values of

and

to make both

and

positive?

2) What is the restriction (i.e. the general rule) for

and

such that no matter what combination of

and

is chosen (but still sums to 100),

and

is always negative?

3) How can you prove your answers to questions 1) and 2) ?

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: How to solve this problem?

Let me clarify what my actual question is. Sorry for not making this clear enough early on.

The question I am asking is this:

1) What restrictions (i.e. the general rule) are needed on

and

so that there are some range of values of

and

to make both

and

positive?

2) What is the restriction (i.e. the general rule) for

and

such that no matter what combination of

and

is chosen (but still sums to 100),

and

is always negative?

3) How can you prove your answers to questions 1) and 2) ?

Well, yes, you should have been more precise as to what you were after – “x[sub]A[/sub], x[sub]B[/sub] are given” is totally different from “x[sub]A[/sub], x[sub]B[/sub] have restrictions imposed on them”, you know.

From my previous working

and after some reshuffling you get

That’s your answer for (1). (Which is why x[sub]A[/sub] = 1.38, x[sub]B[/sub] = 3.1 won’t work.) For (2), follow the same method.