Math Is Fun Forum

Was the "x squared" bit based upon a statistical test? Was it a Chi-Squared Test? Are you sure that this is the correct test?
Or was this based upon linear regression?

There was ONLY SUPPOSED to be EXACTLY ONE illness concerned which ALL the people were known to have.
When you say "illnesses" was that a typing error or did you model it as if you were counting how many illnesses in total
were reported to have occured overal in that month? At the time of writing I have not been able to check how you could
model this. Perhaps Linear Regression can be used if this is the assumption, but that would be your question not mine.

My calculation seemed to suggest about 0.9 % but as you could tell I did NOT trust my answer at all. 3.404% seems right roughly.

That agrees with my rough intuition better than my attempt at a calculation. Was that a linear regression related probability?
Or Chi-Squared? Or Loglinear? Or Multinomial simulation?

Yes I seemed to have "concluded" that we could, but realised that there was something wrong with my attempt.
I thought that using a graphics calculator to fudge a rough solution by trying to "simplify" things was not really going
to give a very accurate answer. I was going to suggest a Loglinear Contigency Table style solution to this, but needed
Genstat to remind me as to whether this was appropriate. I have still got the text book that accompanied the course
which I studied in 2006 on this topic (for which I got a grade 2 pass in a system where grade 1 is best and grade 4 is
just an ordinary pass grade so I was above average, but not exactly amazing at Linear Statistical Modelling).
I think I need a bit of a refresher course on this if I ever do get to use it for something serious, but as yet I have never
had a job which I would need to do this, but you never know what might happen.

I have decided that I accept your answer as correct, but do not know how you reached the answer nor what formal test
you did. You may even have done a simulation of the "exact" situation to see how many times something as significant
or more significant would happen. In which case you could consider that an unbeatable answer.

Thanks.

The cubic equation for dy/dx=0 was:

0 = 12x^3 + 3(k-18)x^2 + (64-12kx) + 8k-48

The DELTA discriminant for whether this cubic equation has:
3 distinct real roots when DELTA > 0
multiple root with all roots real when DELTA = 0
or one real root and two complex conjugate roots when DELTA < 0
(Source: Wikipedia entry for cubic polynomial equations)

Using an algebraic package you can solve this for k where DELTA = 0

So DELTA = 432k^4 + 12096k^3 + 83520k^2 - 345600k - 3998208

I then used a grapical calculator with a polynomial of order 4 solver to get numerical solutions:
Two of them were complex conjugates of (-12.1161977552, +/- 3.62069124665) these are irrelevant for this analysis.
The real solutions are relevant, one of them was Bob's k = 5.9536152399
The other was also mentioned by Bob earlier to less accuracy and I am getting: k = -9.72121972948
These solutions for k indicate that there is a multiple root with all roots real and a graph shows that they have two stationary points
in terms of the original quartic expression involving x and k. One is an inflection and the other a minimum.
You could work out whether there are 3 turning points or 1 turning point for the ranges either side and in between these
if you wanted to give a full analysis of how many turning points you get for all values of real k.
(Perhaps draw a graph on a graphics calulator for the function of DELTA in terms of k and see where it is above zero
and where it is below zero. Then use the wiki quote that I gave above.)
If I am understanding this correctly there is one stationary point inbetween the two (k>-9.7212197... and k<5.9536152...)
and there are three stationary points for k < -9.721297... and for k > 5.9536152...
The exact formula for calculating the roots of k is extremely complicated and I would not like to attempt that one
without a something like Wolfram or another computer algebra package.

I may have joined this discussion too late, but this is my way of doing this question:

The density of water is about 1 g per ml

So 0.0018 ml = 0.0018 g (appox.)

The molar mass of water is: 2x1 + 1x16 = 18 g per mol (approx.)

Then I divided by the molar mass of water to get the number of moles of water:

0.0018 / 18 = 1 x 10^-4 (mol H2O approx.)

Then multiply by Avogadros Constant: 6.022... x 10^23 (atoms or molecules per mol)

1 x 10^-4 x 6.022 x 10^23 = 6.02 x 10^19 (approx.)

Some of the figures I have used are approximate, but since the 0.0018 ml is only
supposed to be approximate I don't think much more accuracy was intended.

Hi

I should point out the following things:
(1) I have not done interest calculations for ages.
(2) The conventions used in the USA may be different to the ones that I studied many years ago.
(3) There are some terms used in the question that I do not know what exactly they mean.

The following link might be helpful:
http://en.wikipedia.org/wiki/Compound_interest

Look at the bit about "monthly mortgage payments" for a mortgage loan calculation.
(I now realise that this will go into some unnecessary depth, but it may still be useful so I have left the link in.)

I am not sure whether this is the same system implied by the question.
I am not sure what "in arrears calculated and charged on monthly rests" in terms of the monthly iteration formula.
However the principal of it may be the same with a few adjustments.

(Actually I have re-read the question and I can now see that the problem is easier than I had thought.)

Yes. I agree. If you let k=0 then z = 1
Obviously (1 + 1)/1 = 2
So this cannot give us 1 when raised to the power of 5.
Why does my algebraic argument not work for k = 0
Usually a fifth root gives us five solutions. (????)

I can see your point about the negative sign as well .....

I am not sure about this because I cannot do the very last bit of trigonometry but this is what I have got so far:

Using a right angled triangle in the Argand diagram:

Using standard trigonometric results we need this in terms of cot(A/2):

I have put in the minus sign in now. Well spotted anonimnystefy.
k = 0  is not a solution I agree Stefy, but why does my answer not work for k = 0 ?

EDIT: I now can see that cot(0) is not possible. Given that cot X = 1/(tan X) if X=0 then we have a division by zero.
Therefore the argument is not valid for this case.

The second attempt, which I originally did not see because I was writing my post at the same time as yours, was correct.
2 x 3 = 6
7 x 3 = 21

So (2/7) = (6/21)

The next step is to do something similar to the fraction: (1/3)

Now that the denominators are the same the addition is easy:

The second one was:

Do you want to have a go at this one yourself ?

Mandy: I have decided to post the first of those questions that I gave you on Friday:
(Q1)

With the statement (h) this to my mind reads "There exists at least one person x who is (happy and a theatre goer and is quiet)".

The statement (3) seems to be the compact form of this.

I will try to help with a bit of LaTeX which can be done using math tags in square brackets, and using things like
\exists
\forall
I am not sure how to do a proper "and" symbol in LaTeX. The same applies to "or" and "not". I have used text for these.

When you have used the ~ symbol does this mean "not" or "a negation of" ?
Try comparing (2) with (a) for example.

I wonder whether this means "is a subset of" or "is a proper subset of". (a "proper subset" of means "smaller subset" of)
Example: Let a set be {1,2,3} a subset would also include the set itself. A proper subset would have to be smaller.
{1,2} is a proper susbset of {1,2,3} (and also a subset).
{1,2,3} is a subset of {1,2,3} (but is not a proper subset).
Try comparing (7) and (g).
Are we saying "not(the set of happy people is a proper subset of the set of theatre goers)" ?

I am not sure about this mainly because I would probably have to actually have to have the text book (or sheet of paper)
that you have quoted that from to be certain. As far as I know this is not a universal concept in terms of the abbreviation.

The equation you have multiplied is done correctly - that is to say you have indeed multiplied by -2.

(As a critical observation: When you write 1/2 x it would be clearer if you wrote (1/2)x because it could be confused with (1/(2x))
which is obviously different and not what you meant. I am being picky here and you knew what you meant because the answer
you then gave was correct.)

The term "R" may mean "row". That is to say it may refer to a row operation - in this case multiplying the entire row by
a factor (in this case -2). If this is the case then I can see what you mean with your answer of -2R.

Whether this is "correct" as such needs a little more information I would think because I am not sure.

Are we defining R to be equal to the entire first row. R = [(1/2) x - 4y = 5]
If this is the case then  -2R = [-x + 8y = -10]

I wonder whether you are supposed to use a subscript notation or something to indicate which row it is refering to.
With matirix calculations there is some notation to do with this. If it is solving simultaneous equations then I am not
sure what the currently taught notation is and it might vary from course to course and differ according to the country
and so on.

However it looks like a sensible answer to me.

I suppose this is designed to teach you to solve 2 simultaneous equations, or possibly 2 by 2 matrix calcuations.
To my mind at least it is the solving a problem bit, rather than what abbreviation should be used, that is the more
important thing.