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ok.

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Find x, y, and z. (Leave answer in simplest radical form)

Hello Guest!

\(y^2=36+x^2\)

\(\underline{z^2=10^2+x^2}\)

\(y^2-z^2=36-100\)

\(\underline{y^2+z^2=16^2}\)

\(2y^2=192\\ y^2=96\\ \color{blue}y=4\sqrt{6}=9.798\)

\(z^2=16^2-y^2\\ z^2=16^2-96\\ z^2=160\\ \color{blue}z=4\sqrt{10}=12.649\)

\(x^2=z^2-10^2\\ x^2=160-100\\ x^2=60\\ \color{blue}x=2\sqrt{15}=7,746\)

  !

Hello all, just a few resources for using LaTeX.

I found a great resource on AoPS about LaTeX. It's not a gigantic read, so I suggest that if you want to learn LaTeX to a deeper extent to look at this webpage. It also covers aligning equations.

https://artofproblemsolving.com/wiki/index.php?title=LaTeX:LaTeX_on_AoPS

If you want to create LaTeX Tables, use this webpage to generate one.

https://www.tablesgenerator.com/#

The only problem is that webpage doesn't allow math input for the tables.

You can also learn to make them yourself in the below link.

https://www.overleaf.com/learn/latex/tables

If you don't know how to do some LaTeX symbols, that's no problem. You can just directly search them up online, and there will definitely be some great sources.

If you are making complicated, hard-to-read equations in LaTeX, then don't spend your time getting confused over the syntax! All you got to do is to type the equation up in the Desmos Graphing Calculator! Copy the result, and then if you paste it, then it will be in LaTeX. All you have to do is to wrap it with dollar signs.

https://www.desmos.com/calculator

-CitrusCornflakes :) $\int_{0}^{\pi}\sin\left(x\right)\ dx\ =\left(-\cos\left(\pi\right)+C\right)-\left(-\cos\left(0\right)+C\right)=2$

A hint:

The number of trailing 0s is the number of factors of 5 in the whole expression. So if you can work out an approach to find out all the factors of 5, then you've solved it.

-CitrusCornflakes :)

We know all factorials 10 and afterwards end in 00 (as there are two factors of 5 and enough factors of 2s), so it is really 5! which counts. This ends in 20, so the last two digits of the described following sum is clearly 20.

Answer:20

-CitrusCornflakes :)

@above, simply functional evaluation. For p(1,-1), then the case is 'otherwise' so we have 3*1-1=2. For p(-5,-2) we have case 2, so then -5-2(-2) equals -1. The outer function then has arguments 2 and -1, which is in the case 'otherwise' so we have 6-1=5.

So the final functional output is 5.

So we have side CD and side BD. We want to find side BC first to work out tangent. Just use pythagoream theorem, so BC=$\sqrt{5}$. Now, you use that Tangent=Opposite/Adjacent so you have CD/BC which is $\frac{\sqrt{14}}{\sqrt{5}}$.

:)

I believe it was a mathcounts question. It was on a alcumus quiz and I got the answer wrong.

XY ≈ 17.685

https://web2.0calc.com/questions/2-probs-help-quick

https://web2.0calc.com/questions/homework-help-pls

Triangles PQA and CRD are similar!!!

AQ / AP = DR / DC

1 / 2 = DR / 5

DR = 5/2

The sum is 40500.

The total weight is 24 pounds.

The question is about a "perfect square that can be represented as the difference of two consecutive PERFECT CUBES" !?

I graphed it, and found 2 solutions.

The domain is (-inf,-8) U (-8,inf).

What would x minus x^2 be?     

If you want a numerical answer, we'd have to know the value of x. 

If you just want another way to say it, well, there's  x • (1 – x)   

_.

Grrr, I said it wrong.  I didn't mean 13 slices have topping.  I meant there were 13 applications of topping to 12 slices.   

_.

12 slices of pizza, and 13 slices have topping.  That means there's only 1 slice that has both. 

Five angles of a hexagon measure 119°, 129°, 140°, 129°, and 95°. What is the measure of the sixth angle?   x 

The sum of the interior angles of any polygon with n sides can be calculated as 180^o • (n – 2)  

The sum of the interior angles of a hexagon is 180^o • (6 – 2)  =  720^o    That's for all six angles.  

The other five angles 119^o + 129^o + 140^o + 129^o + 95^o  total  612^o   

So by subtraction,  720^o – 612^o  =  108^o  is the size of the sixth angle.  

_.

BOLD FACE,     UNDERLINING    AND     ITALICS

What is the domain of the function
f(t) = 1/((1 - t)^2 - (1 + t)^2)?

Hello Guest!

\(\mathbb D=\mathbb R\ |\ t\neq 0\)

  !

Thanks Alan,

Hi Jurimagic,

Perhaps you have not looked at what Alan has done with a fresh eye.

This is ultimately a difference of 2 squares factorization.

Incidentally, I know, and expect Alan knows, that you are always grateful for the help we give you.

That is why, if I am on and I see your question, I will always answer quickly.   

\(p^2-9s^2+6pq+9q^2\\ =p^2+6pq+9q^2-9s^2\\ =(p^2+6pq+9q^2) - (9s^2)\\ =(p+3q)(p+3q) - (9s^2)\\ =(p+3q)^2 - (9s^2)\\ \text{This is the difference of 2 squares}\\ =(p+3q\;\textcolor{red}{\textbf{-}}\;3s)(p+3q\;\textcolor{red}{\textbf{+}}3s)\)

LaTex Coding:

p^2-9s^2+6pq+9q^2\\
=p^2+6pq+9q^2-9s^2\\
=(p^2+6pq+9q^2) - (9s^2)\\
=(p+3q)(p+3q) - (9s^2)\\
=(p+3q)^2 - (9s^2)\\
\text{This is the difference of 2 squares}\\
=(p+3q\;\textcolor{red}{\textbf{-}}\;3s)(p+3q\;\textcolor{red}{\textbf{+}}3s)

or...

BD = √73

CE / 8 = 3 / √73

CE = 24 / √73