Melody

What are the multiples of 8?

You can start it surely. What are the factors of 300.

In other words, what goes into 300?

x^3-3x+1=0

Let y = x^3-3x+1

y'=3x^2-3

y''=6x

Find turning points    y'=0

3x^2-3=0

3(x^2-1)=0

3(x-1)(x+1)=0

x=1 and x=-1

When x=1 y=1-3+1=-1

When x=-1 y=-1+3+1=3

So there is 1 root between 1 and -1

When x=0 y=0-0+1=1

So there is 1 root between 0 and 1

Consider concavity

y''=6x  

If    x>0    y''>0   concave up

thereforethere will be another root that is greater than 1

If    x<0   y'' < 0 concave down

Therefore there will be another root that is less than -1

There can only be a maximum of 3 roots because concavity onle chatnges once.

So there are 2 positive roots and 1 negative root.

I am not sure if this is what you are expected to produce but it works for me.   

\(16^{1/4}\qquad means \qquad \sqrt[4]{16}\)

in others words find a number x such that

\(x\times x\times x\times x=16 \)

I can see that the answer is 2  (-2 also works but it is conventional to give the positive answer.)

2*2*2*2=16   

so

\(16^{1/4}\quad = \quad \sqrt[4]{16}\quad = \qquad 2\)

You could of course get the answer from any scientific calculator.      :)

Mon 9/11/15

2)    Unusual simultaneous equation     Thanks Melody and Heureka.

        http://web2.0calc.com/questions/hello-this-is-juriemagic#r2

3)   Trig simplification       Thanks Melody and Maximillian

      http://web2.0calc.com/questions/cos-2x-cos-4x#r2

@@ What is Happening?  [Wrap4]    Sun 8/11/15   Sydney, Australia Time 9:26pm   ♪ ♫

Hi all,

Thanks Rom, CPhill, Cars456 and Alan for you great answers today. 

Repeat request:

There is a post below where I have invited people to discuss their views of the forum.  All sensible thoughts will be greatly appreciated.

Interest Posts:

If you ask or answer an interesting question, you can private message the address to me (with copy and paste) and I will include it.  Of course only members are able to do this.  I quite likely will not see it if you do not show me.  

1)   Have your say on the new forum.

Plot the points on a graph.

You can use desmos graphing calc if you want to.

The exponential function 

N = 3.93 × 1.34 ^d gives the approximate U.S. population, in millions, d decades after 1790. (The formula is valid only up to 1860.)

Find a formula that gives the U.S. population c centuries after 1790. (Assume that the original formula is valid over several centuries.)

\(3.93 × 1.34 ^d=3.93\times a^{0.1d}\\ 1.34 ^d=a^{0.1d}\\ log(1.34 ^d)=log(a^{0.1d})\\ d*log(1.34)=0.1d*log(a)\\ log(1.34)=0.1*log(a)\\ 10log(1.34)=log(a)\\ log(a)=10log(1.34)\\ 10^{log(a)}=10^{10*log(1.34)}\\ a=10^{10*log(1.34)}\\ a\approx 18.6659\qquad (4dp)\)

N = 3.93 × 1.34 ^d

N = 3.93 × 18.6659 ^C                         C for century

check after 1/2 cnetury

N = 3.93 × 1.34 ^5                 =16.979

N = 3.93 × 18.6659 ^0.5       = 16.979

after 5 centuries

N = 3.93 × 1.34 ^50              = 8 904 970

N = 3.93 × 18.6659 ^5          = 8 905 067         difference due to rounding

A bag contains 6 red marbles, 6 blue marbles, 10 white marbles and 7 yellow marbles. You are asked to draw 4 marbles from the bag without replacement. In how many ways can you draw two blue marbles? what formula do you use for questions like these

I am going to play with this question a little

If ALL the b***s are different. I mean, you have blue 1 blue 2 etc then

Number of ways of choosing 2 blue from 6 is 6C2 = 15

Number of ways of choosing 2 from the others 23C2 = 253

Total = 15*253 = 3795 ways

The number of ways that 4 marbles can be drawn with no restrictions is 29C4 = 23751

So the probablility of drawing 2 blue is    3795 / 23751 = 0.1598    (4 dp)

This is the way it must be done if you are looking at probablilities

BUT you are not concerned with probablilities and I think below is what the question may really be asking.

Lets see,

6 red marbles, 6 blue marbles, 10 white marbles and 7 yellow marbles

you could have 

So there are 6 different combinations that could include 2 blue b***s BUT they are NOT all equally likely to be drawn.  

divide 45km in the ratio 42:58

42+58 = 100

The portions will be      42/100  * 45 =  0.42 * 45  =  18.9 km

and

58/100 * 45    0.58 * 45  =   26.1 km

check

18.9+26.1 = 45   good