Thankyou anonymous for providing the puzzle and thanks AzizHusein for providing the clever solution. 1/8/14

Does

y Σ x X ^

mean anything to anyone?

I'm impressed too.  I'm going to put this one into our 'puzzles' sticky thread!

Thanks to both of you.  

Yes Ninja,  it is easier to take things in if they are well presented.

This is beautiful.  

Since you are only 14 I don't think that you have missed anything just yet! 

Yes that is fine 15threefold.  It is almost the same.

I remember I did not comprehend the relevance of the graph for a long time either.

It is really good when you do get it but your method will always work well and it is almost as easy.  

My answer is the same as Azis's.  Just the presentation is a little different.  

$$cos\frac{17\pi}{6}\\\\
=cos\left(\frac{12\pi}{6}+\frac{5\pi}{6}\right)\\\\
=cos\left(2\pi+\frac{5\pi}{6}\right)\\\\
=cos\left(\frac{5\pi}{6}\right)\\\\
\mbox{This is in the second quad - cos is negative}\\\\
=-cos\left(\frac{\pi}{6}\right)\\\\
=\frac{-\sqrt3}{2}$$

 Melody  

I don't understand some points of your explain but I take your information and solve the question by this way

http://im44.gulfup.com/n3q0s0.jpg 

Focus groups of 12 people are randomly selected to discuss products of the Yummy Company. It is determined that the mean number (per group) who recognize the Yummy brand name is 9.5, and the standard deviation is 0.69. Would it be unusual to randomly select 12 people and find that fewer than 6 recognize the Yummy brand name? 

I think I will just talk about Alan's answer a little.

Mean=9.5

one standard deviation either side of the mean is    

Standard deviation is 0.69

$$\\9.5-0.69\;\;to\;\;9.5+0.69\\
8.81\;\;to\;\;10.19\qquad\\\$$

So approx 67% of the time there will be between 8.8 and 10.2 people who recognise the brand name.

etc

99.7% of the time the number will be between -3 and +3 standard deviation from the mean

100-99.7=0.3 and  a half of 0.3=0.15

Now 3 standard deviation below  the mean is  9.5-3*0.69 = 7.43

there is less than 0.15% chance that the number of people who recognise the product will  be less than 7.43.

So the liklihood that only 6 or less recognise the product is tiny.  It would be extremely unlikely.

This was the question

the difference between twice a number and 8 is twenty more than nine times the number. find the number

now you think that the number is -4

The difference between twice a number and 8.  Well twice the number is -8. 

Umm what is the difference - this is causing all the problem.

So Chris, you think that the difference between -8 and 8 is -8-8=-16  

20 more than 9 times the number is -36+20=-16

So Chris you must think the difference between 6 and 4 is different from the difference between 4 and 6?

Mmm If you have  pile of 6 apples and I have a pile of 4 apples the  the difference the piles is 2 apples.

Isn't it the same answer if I say you have 4 apples and I have 6 apples? Isn'tthe difference still 2 apples?

-----------------------

I also have given more thought to this question.

I started my solution with 

|2x-8| = 9X+20

this lead to the 2 solutions  x=-4 and x= -12/11

These were tested and x=-4 was dismissed.  Now I should have dismissied this answer without the checking.

Anonymous probably already worked through this but I am slack about working out what other people  are telling me so I am trying to work through it myself.  Sorry anonymous, I do not mean any offense by this.

Now since 9x+20 is equal to an absolute then it must be >=0

$$\\9x\ge-20\\
x\ge-20/9\\
x\ge\;-2\frac{2}{9}$$

So this excludes x=-4

Continued here

That depends which integral.

I'll do a very simple one for you.

$$\int 5x^6+\frac{9}{2}x^3 +4 \;dx\\\\
=\frac{5x^{6+1}}{7}+\frac{9x^{3+1}}{2\times4}+4x+c\\\\
=\frac{5x^{7}}{7}+\frac{9x^{4}}{8}+4x+c\\\\$$

$$\frac{4}{523}$$

or

$${\frac{{\mathtt{4}}}{{\mathtt{523}}}} = {\mathtt{0.007\: \!648\: \!183\: \!556\: \!405\: \!4}}$$

This decimal answer is an approximation.  If you had enough digits a recurring fraction would appear.  

Good answer Dragon 

Find out how many standard deviations 6 is from 9.5 and look up the probability, or, better in this case, just look at a graph of the probability density function like below (assuming the distribution is normal).

If you look at the graph, does it seem likely that fewer than 6 people would have heard of Yummy?  Strictly, the graph shouldn't go above 12, so using a normal distribution is an approximation.

31/7/14

Learning Latex Thread for beginners

OK Thanks Heureka   

 {*{20}c}    ?

Hi Melody,

I will answer your question:  the meening of {*{20}c} 

Generally :

*{count}{form}

Example:

{*{20}c}   means  {cccccccccccccccccccc}    "c" twenty times

{*{3}{|r}}  means  {|r|r|r}                        "|r" three times

This is how I would have done it - it is slightly different from yours

This proves that there is more than one way to skin a cat - oh sorry kitty.

I new about using Big but it has never worked fo me when I wanted to use it!

The \left and \right is probably better most of the time because it makes it the correct size, as opposed to a fixed large size.

\left(\bar{x} \pm z\frac{s}{\sqrt{n}}\right)

$$\left(\bar{x} \pm z\frac{s}{\sqrt{n}}\right)$$

Here is a stack of them that you can practice on

Hint:  the flatterned s at the front of each one is an integral sign.

Melody is correct in suggesting that I might should have reversed the signs......however, I couldn't say at that point whether x was negative or not.......all I could say originally is that x ≠ 0....it might have been that x only took on positive values...!!!! Besides, in these types of problems, I like to generally ignore the inequality signs and set up an equality that will give me some intervals to work with - which is what I did. By picking some test points in these intervals, we can determine which values make our original inequality true, and I don't have to necessarily worry about the direction of the inequality sign throughout the problem. This may not be the fastest way to solve such things, but it's the method I learned......

Here's the graph  of both functions......y = 1/x is in red     y = x is in blue

Note that the intervals I have noted in my answer appear to be correct....

Maybe I didn't look at your answer hard enough Chris but I think it has a floor in it.

Sorry Chris I don't think that I read your answer properly.

If you multiply an inequality by x and x is negative then you have to turn the sign around.

Therefore you end up with 2 equations to solve.

It is better to multiply by something that you KNOW is positive.  Squared things are always positive!

Edit:    you cannot divide by 0 so    $$\textcolor[rgb]{1,0,0}{x\ne0}$$

$$\begin{array}{rll}
\frac{1}{x}&\le&x\\\\
x^2\times\frac{1}{x}&\le&x^2 \times x\\\\
x&\le&x^3\\\\
-x^3+x&\le&0\\\\
-x(x^2-1)&\le&0\\\\
-x(x-1)(x+1)&\le&0\\\\
\end{array}$$

Now the easiest way to finish this is with a graph of

  $$y=-x(x-1)(x+1)$$  and see where it intersects with $$y\le0$$

I am going to try and do a quick sketch - I know how to do a proper graph but I want to try and replicate what I would do by hand.

It is a cubic because the highest power of x is 3 and it will finish in the 2nd and 4th quads because the leading coefficient is -1

I can see from the graph that the answer is   (this answer has been very slightly edited)

 $$-1\lex\le x<0 \qquad or \qquad x\ge1\\

[-1,0)\;\cup \; [1,\infty)$$

I think it is right now Chris - Do we have the same answer?

Had to look up a few things, but here it is!

$$\Big(\overline{x} \pm z\frac{s}{\sqrt{n}}\Big)$$

Or, alternatively:

$$\left(\overline{x} \pm z\frac{s}{\sqrt{n}}\right)$$

inverse tan is the same as arc tan

so use atan

We aren't including 0 !!!   That's what the parentheses indicates....if we were including it, we would have had a bracket....the interval is everything from -1 up to, but not including, 0 .......

Alright Ninja,

Here is another one - try reproducing this - not this big!

If you get it out then try enclosing it in big brackets