All Answers 358516 Answers

Thanks Chris,

I had very good maths teachers when I was at school (most of the time).  I am very grateful to them because these formative years are very important to learn the concepts and also for teaching the concepts at a later time.

This building up of graphs was not taught to me at school, perhaps the rudiments of it were but I mostly worked it out for myself.  It is an incredibly useful concept to understand. And it is fun seeing the graphs 'grow' from the germ of a beginning.

Desmos graphing calculator is a fantastic tool to show how the procedure works :)

No.. Heureka and I have got that one nailed!

You should try learning Latex.  It is like everything, the more you know the easier it is to add to your knowledge.  

Getting started is the hardest bit but then it becomes easy.  And you can start with really trivial things here so even getting started is not so hard. :)

It is much easier to makes sense of well presented material - and that usually means LaTex.

just to rephrase it, i got the perfect equation setup

303.92*(10^9)/11.2*(10^6)

10^9=1,000,000,000

10^6=1,000,000

303.92*1,000,000,000=303920000000

11.2*1,000,000=11200000

303920000000/11200000=27135.7142857142857143

answer=27135.7142857142857143

To "b" or not to "b"......that is the question......is it nobler in the minds of men to expandeth answers to infinity through the Queen's and heureka's employment of LaTex, or, should other - more compact - answers serve our  purposes just as well ????

That's pretty neat, Alan.....so......can we assume that....the colder it gets, the more time flies????

 [ Regardless of the climate change....I still gave you 3 points....  !!!!]

what percentage of 100 is 4?

maybe 4%

out of 10=40%

out of 4= 100%

yea this question doesnt make any sense. oh plz don tell me its a rip off of the riddler.

i notice a trend here of -3 points in my comments

i think this meant that the hawaiian pizza wasnt good.

i think ill get a pepperoni or a cheese pizza next

guess not everyones meant for hawaiian

I like that presentation, Melody......it really shows how each component of the function changes the shape/orientation of the curve.....!!!!

Thanks Heureka    

Yep - My turn to make it look difficult.      LOL 

what is b in

  3            2

'''''''''''''=''''''''''

3b+4      b - 4

$$\small{
\begin{array}{rll}
\dfrac{3}{3b+4} &=& \dfrac{2}{b-4} \\\\
\dfrac{3b+4}{3} &=& \dfrac{b-4}{2} \\\\
b + \dfrac43 &=& \dfrac{b}{2} - 2 \\\\
\dfrac{b}{2} &=& - 2 - \dfrac43 \\\\
\dfrac{b}{2} &=& -\dfrac{10}3 \\\\
b &=& -\dfrac{20}{3}\\\\
b &=& -6\dfrac{2}{3}\\\\
\end{array}
}$$

@@ End of Day Wrap  Thurs 2/7/15   Sydney, Australia Time   9:20pm  ♪ ♫

Good evening,     

Today our exceptional answers were provided by Sir-Emo-Chappington CPhill, Alan Radix, WizBoy, Mellie, Heureka, Oscarcampos02, MathsGod1 and TitaniumRome.   Thank you  

$$\begin{array}{rll}
\frac{2}{x^2-5x+6}+\frac{5}{x^2-4x+3}&=&\frac{6}{x^2-3x+2}\\\\
\frac{2}{(x-3)(x-2)}+\frac{5}{(x-3)(x-1)}&=&\frac{6}{(x-2)(x-1)}\\\\
(x-3)(x-2)(x-1)\left(\frac{2}{(x-3)(x-2)}+\frac{5}{(x-3)(x-1)}\right)&=&(x-3)(x-2)(x-1)\left(\frac{6}{(x-2)(x-1)}\right)\\\\
2(x-1)+5(x-2)&=&6(x-3)\\\\
2x-2+5x-10&=&6x-18\\\\
7x-12&=&6x-18\\\\
1x&=&-6\\\\
x&=&-6
\end{array}$$

Do you mean

$$\begin{array}{rll}
\frac{3}{3b+4}&=&\frac{2}{b-4}\\\\
(b-4)(3b+4)\times\frac{3}{3b+4}&=&\frac{2}{b-4}\times(b-4)(3b+4)\\\\
(b-4)\times\frac{3}{1}&=&\frac{2}{1}\times(3b+4)\\\\
3(b-4)&=&2(3b+4)\\\\
3b-12&=&6b+8\\\\
3b-12-3b&=&6b+8-3b\\\\
-12&=&8+3b\\\\
-12-8&=&8+3b-8\\\\
-20&=&3b\\\\
b&=&\frac{-20}{3}\\\\
b&=&-6\frac{2}{3}\\\\
\end{array}$$

check

LHS =    $${\frac{{\mathtt{3}}}{\left({\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{\,-\,}}\left({\frac{{\mathtt{20}}}{{\mathtt{3}}}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{4}}\right)}} = {\mathtt{\,-\,}}{\frac{{\mathtt{3}}}{{\mathtt{16}}}} = -{\mathtt{0.187\: \!5}}$$

RHS =      $${\frac{{\mathtt{2}}}{\left({\mathtt{\,-\,}}\left({\frac{{\mathtt{20}}}{{\mathtt{3}}}}\right){\mathtt{\,-\,}}{\mathtt{4}}\right)}} = {\mathtt{\,-\,}}{\frac{{\mathtt{3}}}{{\mathtt{16}}}} = -{\mathtt{0.187\: \!5}}$$

Fri 3/7/15

If you would like to comment on other site issues please do so on the Lantern Thread.  Thank you.    

Interest Posts:

FTJ means 'For the juniors' 

Your question does not make sense.

Perhaps you could rephrase it?

You are on a roll today Chris.  Your logic is irrefutable      

(That's what I think anyway)      LOL

That is great logic Chris.  

It is so simple and so logical but I would not have thought of it   

That is really good logic I think Chris    

I forgot the 'divide by 49' one when i did it.                

Thanks Chris,

I am going to do it too    

$$\\40^8\times75^{18}\\\\
=10^8\times 4^8\times (\frac{3}{4}\times 100)^{18}\\\\
=10^8\times 4^8\times \frac{3^{18}}{4^{18}}\times 100^{18}\\\\
=10^8\times \frac{3^{18}}{4^{10}}\times 100^{10}\times 100^{8}\\\\
=10^8\times 100^{8}\times 3^{18}\times 25^{10}\\\\
=10^8\times 10^{16}\times 3^{18}\times 25^{8}\\\\
=10^{24}\times 3^{18}\times 25^{8}\\\\
$25 to the 8 will end in a 5$\\
$3 to the power of anything does not end in a 0 $\\
$The last digits of powers of 3 are 3,9,7,1,3,9,$\\
$There are 4 numbers in the pattern. 18=2mod4$\\
$Hence 3 to the 18 ends in a 9$\\
so\\
3^{18}\times 25^{8} $ will end in a 5$\\\\$$

So there will be 24 trailing zeros - just like CPhill said   

$${\frac{{\mathtt{33}}{\mathtt{\,\times\,}}{\mathtt{2}}}{{\mathtt{6}}}} = {\mathtt{11}}$$

$${\frac{{\mathtt{2}}}{{\mathtt{6}}}} = {\frac{{\mathtt{1}}}{{\mathtt{3}}}} = {\mathtt{0.333\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

33 : 3 = 11

You are correct Chris.  

In fact there are 3 pairs of integers that sum to 1001 and and have the largest gcd of 143, namely: (143, 858), (286, 715), (429, 752)   (All multiples of 143).

.

Thank you anon - I really like it that you have gotten involved.   

Yes, I think Chris is right

$${factor}{\left({\mathtt{144}}\right)} = {{\mathtt{2}}}^{{\mathtt{4}}}{\mathtt{\,\times\,}}{{\mathtt{3}}}^{{\mathtt{2}}}$$

1^1=1

2^2=4

3^2=9

(2*2)^2=16

(2*3)^2=36

(2^2*3)^2=(4*3)^2=144

.

y=-1/2sin[4(x+pi/4)]+1

You can build these up bit by bit.  You DO need to know what y=sinx looks like

y=sinx  has a frequency of 2pi    

y=sin4x  Will have a frequency of  2pi/4

The point (0,0) on the curve sin4x will be moved to (-pi/4,0) on sin(4(x+pi/4)

That is, the phase shift will be pi/4 to the left.

y=sinx has an amplidude of 1. It oscilates between y=-1 and y=1    

y=1/2 sinx has an amplitude of 1/2

and y=-1/2sin x will reflect it across the x axis (turn it upside down)

 The plus one ar the end will lift the whole curve up by one.

So instead of it oscillating between y=-0.5 and 0.5  it will oscillate between y=0.5 and 1.5

I can show you these steps on a Desmos graph

I will only show the sine graph at the beginning then I will step through the building of the final graph one step at the time.

Each graph can be displayed (or made invisable again) by clicking the circle in front of its formula  

You made that look REALLY hard Heureka   LOL   :))