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How did you find the reciprocal  of   [ 1 + 3^(1/3) + 3^(2/3) ]   ????

I took the reciprocal of the denominator for which I got: 1/2 (3^(1/3) - 1) =3^(1/3)/2 - 1/2. Lastly, I multiplied this by the first term which gave me the result above.

We have a right triangle with the legs   =  (2120 - 1650)  = 470 m  and  950 m

And the direct distance between them  = the hypotenuse =   √ ( 470^2 + 950^2)  ≈  1059.9 m

The smallest angle would be: sin^-1(3/5) =~36.87 degrees.

Yep, guest......that looks to be correct....good job  !!!

BTW -  the denominator  is   ( [ 1 + 3^(1/3) + 3^(2/3)] [ 3^(1/3)/2 - 1/2])  which simplifies to 1

What tipped you off  ???

I got it! =2 3^(2/3) (3^(1/3)/2 - 1/2)=2[3/2 - 3^(2/3) /2]=[6/2 - 3^(2/3) =3 - 3^(2/3) !!

1)sin127 cos82- cos127 sin82

Notice  that  this actually simplifies to  sin (127 - 82)  =  sin (45) =  1 / √2

2)sin^2(20)+ cos^2(20)   =   1         [ using the identity  sin^2(theta) + cos^2(theta)  = 1 ]

3) sin^4alpha+ sin^2alpha• cos^2alpha+ cos^2alpha=1

sin^4alpha  + sin^2theta  (1 - sin^2theta)  + cos^2alpha

sin^4alpha  + sin^2alpha  - sin^4alpha  + cos^2 alpha =

sin^2alpha   +  cos^2alpha   = 1          and this is true

4) sin2x-sinx=0    

2sinxcosx - sinx  = 0      factor

sinx ( 2cosx - 1)  = 0         set each factor to 0 and solve

sin x  = 0   at   0 + pi *n      where n is an integer

And for the other factor

2cosx  - 1  = 0       add 1 to both sides

2cosx  = 1            divide both sides by 2

oos x   = 1/2      

And this happens  at  pi/3 + 2pi*n    and   at    5pi/3 + 2pi*n    for some integer, n

 no one wants to help me so I guess I'll answer it myself (forever alone) "they had only one pair of trunks"

Vertice is any node in the graph.

Parent is a vertice that has at least one path out of it leading to another vertice.

Leaf is a vertice that is not a parent.

let P be the number of parents and L the number of leafs. The folks that didn't forward the email are the leafs.

V=1501 = 2P+ 1

P=750

V=1501 = P + L

L = 751

I still need to think about why this always works ...

\(-2.\dot6=-2\frac{2}{3}\)

(3⁵)² / 3^-2    =

3¹⁰  / 3^-2        and using    a^m / aⁿ    =   a ^m-n,  we have

3^{10 - (-2)}    =     3^{10 + 2}    =   3¹²   

Yes I saw that too Chris and I also have do not know how they did that. 

Here is a definition that I found:

What Is Technology? Technology is a body of knowledge devoted to creating tools, processing actions and extracting of  materials. The term ‘Technology” is wide and everyone has their own way of understanding the meaning of technology. We use technology to accomplish various tasks in our daily lives, in brief; we can describe technology as products, processes or organizations. We use technology to extend our abilities, and that makes people as the most important part of any technological system.

you can do it yourself 

What is the equation of a parabola with (−2, 4)  as its focus and y = 6 as its directrix?

Well it is concave down with a focal length of 1 and the vertex at (-2,5)

so

\((x-h)^2=4a(y-k)\\ (x+2)^2=4*-1*(y-5)\\ (x+2)^2=-4(y-5)\\\)

Hey..I think I can make up one that only requires basic trig and algebra knowledge 

Once upon a time there was a boy named Chris. He liked to eat potato chips and watch football on TV. One day, Chris decided that he needed a place to store all his potato chips. So, he set out to make a gigantic silo to hold all the chips.

The bottom was to be shaped like a cylinder, and the top shaped like a cone. The entire thing, from the lowest point to the highest point, was to be 100 feet tall.

The diameter of the cylinder, as well as the cone at the top, will be 30 feet. The angle formed inside the silo by the wall of the cylinder and the wall of the cone will be 110º.

What is volume of the silo?

Here's roughly how it looked like:  

2*9^(1/3)  / [ 1 + 3^(1/3) + 9^(1/3) ]  =

2*3^(2/3) /  [ 1 + 3^(1/3) + 3^(2/3) ] 

WolframAlpha shows that this can actually be simplified to  

3  - 3^(2/3)

How they got there, I don't know...I tried to use a conjugate, but didn't get anywhere.....maybe I didn't follow it through far enough???....maybe some other mathematician knows how to simplify this step-by-step....!!!!

 simplify | (2 9^(1/3))/(1 + 3^(1/3) + 9^(1/3)) =3 - 3^(2/3)*

* My Mathematica 11 Home Package came up with that answer but will not give details or steps as to how they got that answer!!

You want a challenging 3D geometry question, but you have only basic trig and algebra knowledge and no graphing knowledge.

That’s like saying you want to climb Mt Everest or K2, but you only have two dinky pitons, no rope, and you can’t climb the molehills in your backyard, even when you can find them. 

\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}.

\(\frac{2\sqrt[3]9}{1 + \sqrt[3]3 + \sqrt[3]9}\\\)

I do not think this can be simplified.

Since the denominators in your sequence follow a pattern of: 6, 24, 60, 120, 210, 336...etc. and continue at this pattern:n^3+3n^2+2n......to infinity, the reciprocal of all terms converging to 1/4. Therefore, your sequence will converge to: 2[1/4] =1/2.

Technology has been around for ever. When the first cave man chipped at a stone to make a sharp blade, that was technology. 

Yep! There is a " ! " button right next to the " log " button.
Or, you can just type an exclamation point after the number you want "factorialized" (?) 

On this calculator, you can type them in like this:

numerator / denominator

For example...

To do:   \(\frac45\)

Type: 4/5

If the numerator or denominator contains another operation, make sure to put parenthesees around it.

Also, on the right end of the input bar, you can click "input helpers" and select "fractions."

Thanks, Melody!

#When Ii wrote this, I was in the public library on another device.