All Answers 358089 Answers

\(\quad \displaystyle\sum^{12}_{n=5}n^2\\ =5^2 + 6^2 + 7^2 +\cdots + 12^2\\ = 620\)

Let a and b be those 2 numbers.

\(\begin{cases}a+b=7\\ab=2\end{cases}\\ a + \dfrac{2}{a} = 7\\ a^2 -7a + 2 = 0\\ a = \dfrac{7 \pm \sqrt{45}}{2}\\ a = \dfrac{7 +3\sqrt5}{2} \text{ OR }a = \dfrac{7-3\sqrt5}{2}\\ \text{When }a = \dfrac{7+3\sqrt5}{2},b=\dfrac{7-3\sqrt5}{2}\\ \text{When }a = \dfrac{7-3\sqrt5}{2},b=\dfrac{7+3\sqrt5}{2}\\ \text{The solutions are }(a,b) = \left(\dfrac{7-3\sqrt5}{2},\dfrac{7+3\sqrt5}{2}\right)\text{ or }(a,b) = \left(\dfrac{7+3\sqrt5}{2},\dfrac{7-3\sqrt5}{2}\right)\)

He will be 2(3/4 - 9/16 + 27/64 - 81/256 + ... ) meters from his home finally.

This is a G.S. with first term 3/2 and common ratio -3/4. Sum of the G.S. (i.e. the final distance from home) is \(2\left(\dfrac{\dfrac{3}{4}}{1-\left(\dfrac{-3}{4}\right)} \right) = \dfrac{6}{7}\).

So A = 6/7, B = 2 - 6/7 = 8/7.

|A - B| = |(6-8)/7| = \(\dfrac{2}{7}\).

What number?  Can I use a calculator?

Thanks guest, your method obviously works  

I like my method because it is tidier. I mean I don't have everything over my dsektop.  

But I guess if you wanted you could dedicate a file to these types of saves and that way would make your method quite tidy.

Your method is certainly simpler in that no downlad program is needed.  

Please help

I can't give you a detailed solution, but i give you the solution by wolfram alpha:

WolframAlpha: solve  y^2-zx = -103, z^2-xy = 22, x^2-yz > 81 over the positive integers

So \(x^2-yz = 14^2-3\cdot 8 = 172\)

How do i get this \( x^2-yz > 81\) ?

\(\begin{array}{|rcll|} \hline && (x^2-yz) + (y^2-zx) + (z^2-xy) \\ &=& x^2+y^2+z^2-yz-zx-xy \\\\ &=& \dfrac{1}{2}(2x^2+2y^2+2z^2-2yz-2zx-2xy) \\\\ &=& \dfrac{1}{2}\left( (x^2-2xy+y^2) + (y^2-2yz+z^2) + (z^2-2zx+x^2) \right) \\ &=& \dfrac{1}{2}\left( (x-y)^2 + (y-z)^2 + (z-x)^2 \right) \quad | \quad \text{this is always positive} \\\\ (x^2-yz) + (y^2-zx) + (z^2-xy) &>& 0 \\ (x^2-yz) -103 + 22 &>& 0 \\ (x^2-yz) -81 &>& 0 \\ \mathbf{x^2-yz} & \mathbf{>} & \mathbf{81} \\ \hline \end{array}\)

I save it first to my desktop then click on 'Image' Upload, then choose the file on my dekstop and submit it. 

As a ratio

6 1/3 / (8 1/2)   =    x / (4 1/4)

4 1/4   *    6 1/3 /(8 1/2)  =    3 1/6 ft redwood

Here is one of cupcakes's images

I think....

To do that the file image would have to be uploaded somewhere already and only certain file formats would be accepted.

Is this correct?

I just tried to put in one of cupcake's pics your way CalculatorUser.

I thought it worked but the pic disappears when I go to 'show preview.'   

All I do is right click on the image I have and I click copy image address. Then when inserting the image, I just copy the address down.

For some reason, uploading the images on a Chromebook doesn't work. I think the Web2.0 Admin should fix that.

If the image can't be right clicked, the Chromebook actually has an automatic feature that captures any part of the screen just like Gyazo, so that's how I insert pictures.

The wavelength here is    \(\pi \)   and that is the only transformation from the parent function of   y=cos x

If y= cos nx   then the wavelength will be     \(\frac{2\pi}{n}\)

so     

\(\frac{2\pi}{n}=\pi\\ \frac{2}{n}=1\\ n=2\)

So it must be      \(y=cos(2x)\)

Check by using Desmos graphing calculator.

Desmos is another tool that is very easy to use and very helpful to you. 

Thank you! :)

It is a sin wave   with origin 0

    it is shifted UP by 2

 sin x  + 2

Take the square root of 2007 =~44.8 - round it up to the next whole number, or 45.

45^2 = 2025 - 2007 = 18 - years.

8t^2 <= 3-10t

8t^2 + 10t - 3  <=0    Quadratic formula results in zeroes at 1/4 and - 3/2

       this is an UPWARD opening parabola, so the values less than zero are between these two zeroes    [-3/2,1/4]

You didn't specify a starting point, so I'll choose zero. 

What I get is

  0.00

  0.75

  0.75

  1.50

  2.25

  3.75

  6.00

  9.75

15.75

25.50

total = 66

I'm not sure if you are going to like how I solved the problem, but I solved it anyways.

We already know that \(40^2\) is 1600 and that \(50^2\) is 2500, that means the answer is between 40 and 50. Seeing that 2000 (2007 rounds to 2000) is closer to 1600, we try if \(44^2\) will fit the answer. \(44^2\) apparently is 1936, so the next possible solution is 45. Seeing that \(45^2\)is 2025, we can know that \(\boxed{18} \) years after 2007 will be the answer

If somebody finds a better way to solve this, please post it.

Electric Pavlov's answer except in clearer format

\((xy)*(\frac{1}{x}*\frac{1}{y})=\frac{xy}{xy}=\boxed{1}\)

Think about getting a cube of clay and molding it into a ball....the ball will be taller.

Cube volume = h x h x h              10 in ^3 = h^3    h = 2.15 inches tall

Sphere volume = 4/3 pi r^3 = 10 results in r = 1.33      Diameter = 2r = 2.67 inches tall

It's the ratio of the circumference of a circle to its diameter.

If the diameter of a circle is 1, then its circumference is 3.14•••

800 + 48(519.63)    - 22500 = $3242.24 INTEREST

One          x  * y   *   1/x * 1/y=    x * 1/x   * y * 1/y   = ONE