Melody

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.  

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

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