help-13

[crimson solstice]

and you do this over them

the output of each encoder is just a vector

thus, it is a vector space

what would you call the resultant vector space in relation to the 2 previous vector spaces?

I was thinking something along the lines of conjuntive vector space idk if that's an actual term

maybe someone has read out the proper term for that resultant vector space

[urban coral]

think the channel's not reopened yet

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[chrome elk]

How has this channel not moved to available or been hidden yet

.reopen

Interesting

.close

Weird

[normal cipher]

.close

[wicked mantle]

.reopen

[normal cipher]

okay

[wicked mantle]

.solved

[normal cipher]

bot issues

its been half a day

[frank minnow]

huh

[nova snow]

hello

.claim

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[native heath]

.close

.claim

hella hi

hello hi

.reopen

.open

.solved

.done

hmmm

insane

useless bot

[cedar kiln]

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[torn marsh]

what did I do wrong here?

[torn marsh]

Maybe the phase shift

the period starts at -1 right?

[topaz pewter]

you've only drawn y= sin(x) here

[torn marsh]

all my increments are in radians

but -1

I think its radian too but its inconsistent to the others

[urban coral]

inconsistent to what

[torn marsh]

all icrements express a Pi

[quartz coral]

you started the period at 0

instead of -1

[torn marsh]

yea thats because its 0

but -1 is a diff story

[urban coral]

which is what people are trying to tell you here, sir

[quartz coral]

okay so

if you have a trig function

lets say

y = asin(bx+c) + d

a is the amplitude

[torn marsh]

mhm

[quartz coral]

it usually stretches the graph

b controls the period*

and how often it repeats

while c

[torn marsh]

yep

[quartz coral]

which is in your case

[torn marsh]

is the phase shift

[quartz coral]

thats a horizontal shift

vertical shift is d

[torn marsh]

d is the vertical shift

[quartz coral]

yes

[torn marsh]

yea

I understand that

[torn marsh]

wdym

[urban coral]

?

I don't think that sentence is very ambiguous...?

[torn marsh]

I mean what were thy saying

[urban coral]

well I mean look at the message right above the message in this screenshot

[quartz coral]

it may seem confusing because

it's sin(x+1)

and c is +1

but that c tells you how much you have to move it to the left

[torn marsh]

c is -1

right

not +1

[quartz coral]

in your photo it's +1

The SHIFT is -1

[torn marsh]

nvm c is 1, x is -1 ok

[quartz coral]

yes

[torn marsh]

so the period ends at 2pi -1

this is what I did

Im pre sure its wrong

[topaz pewter]

In your first graph you had a zero at pi

So you need to shift it by -1

(pi-3)/2 isn't a zero

[torn marsh]

hm how do I do that

[topaz pewter]

So your original graph is y = sin(x)

Then you need to shift everything

so you look at each zero of sin(x) and you change it by -1

[cedar kiln]

@torn marsh Has your question been resolved?

[torn marsh]

Ok i think i understand now thanks

.close

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[mint depot]

I just getting started learning this topic I think is called set theory in englisch. and my brain doesn't get the red part and I think thats something that only a person can explain to me.

[mint depot]

thanks in advance

[jade blaze]

u do know what N means right?

[floral arrow]

It just means n/2 must be a natural to be included in C

[mint depot]

so if its 3 or 5 its false or something like that?

[clever venture]

yea

so it's not on the set

think of what numbers could apply here and if there's a pattern

[jade blaze]

negative numbers too

[vernal flume]

Natural numbers

[mint depot]

right almost forgot

[jade blaze]

in false cases

[clever venture]

he's saying that negative numbers are also false

because it's $\in \mathbb{N}$

[digital bear]

So it has to be both even and equal or smaller than 8 for it to be C?

[wraith dagger]

MarcoMa210

[digital bear]

Not sure though

[clever venture]

it's exactly that

[digital bear]

Not equal my bad natural

So nu negatives and 0

[clever venture]

controversial take there

[digital bear]

Wait 0 is natural right?

So yeah no negatives

[clever venture]

depends on what the textbook says idk

@mint depot does 0 belong in the natural numbers?

[cobalt iron]

0 not natural no.

[mint depot]

my brain cant process how its written. I think I have problem to read the definition

[mint depot]

I think not

.close

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[crimson sedge]

is the ans to both the parts of 3/1/37 yes

[crimson sedge]

?

i proved u may reach to (a,b+k) where k is any itneger from (a,b)

so we reach (0,2025)

now we reach( 1,2025) using the claim we used above

now we reach (1,1/2025)

and then we reach (2025,1.2025)

*1/2025

for the 2nd part i again reach (0,2025) then same procedure

now the proof of my claim is as:
3

(a,b) to (a+1,b+1)

then to (-a-1,b+1)

shit i diproved my claim

my claim here now changes to u may reach (a,b+k) where k is an even integer

so im not sure abt first but second shall be yes

100%

any1 help me verify my proof and 1st part

or maybe we may prove we can prove we reach (x,1/x) everytime or some conditional x from (0,1)

[cedar kiln]

@crimson sedge Has your question been resolved?

[ancient lodge]

No, as per the solutions

https://files.usamts.org/Solutions_37_1.pdf

[crimson sedge]

thnx i was searching for these

.close

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[wary forge]

I have been able to disprove for alpha <= to 0, but I have not been able to find a usable bound for |f(x)-f(y)| to prove that it is uniformly continuous for alpha > 0

[wary forge]

I tried to use the fact that |f(x)|<|x^alpha| but my triangle inequality would be going in the wrong direction for me to apply it to |f(x)-f(y)|

[neat dune]

try extending to a continuous function on [0,1]

[wary forge]

I don't know how to do that

[sand cradle]

this! By Cantor, a continuous function on a closed + bounded set is uniformly continuous

(on a compact set even. Proof: contradiction)

[neat dune]

well the 1 part is pretty trivial, it's the same function with x = 1 plugged in

[wary forge]

is this just taking the right-hand limit?

for the function at 0?

[neat dune]

as for x = 0, maybe try it with the limit of xsin(1/x) as x goes to 0 for example

yeah

[wary forge]

Alright, I will see what I can do

[sand cradle]

You expect it to go to 0 (just take alpha = 1, 2, .., sin is bounded) so you can also just throw absolute values onto it, then bound above and show that bound goes to 0 for appropriate alpha (then you save yourself of squeeze theorem)

[wary forge]

ok so the |f(x)| <= |x^alpha| is useful, I just needed to look at it for squeeze theorem

[sand cradle]

I meant like this, you kinda save yourself from taking limits on both sides as you would with -x^alpha <= f(x) <= x^alpha, and so you don't really need squeeze theorem

You know 0 <= |f(x)| and when that holds for all x, it must also hold for the limit

So that's not directly squeeze theorem there

[sand cradle]

Now you can determine for which alpha that goes to 0

[wary forge]

Ok, I think I did it

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[silver rover]

Hi can someone please help me with this problem? I understand its physics but hopefully someone here can maybe guide me a little? I've attached a picture of the problem and my work. I'm not getting the answer from the answer key, but I feel like my process is right

[silver rover]

(sorry for the orientation of the photo of my work)

[cedar kiln]

@silver rover Has your question been resolved?

[urban coral]

,rccw

[cedar kiln]

@silver rover Has your question been resolved?

[tough halo]

mommy that's physics

[tough halo]

hi mommy so my ai says it's 45 seconds

[silver rover]

Which AI

[tough halo]

I can't dislose that

[silver rover]

What is the process?

[gloomy vigil]

Alright, I will see what I can do

[cedar kiln]

@silver rover Has your question been resolved?

[sick tusk]

Hey I hope you are still there, anyways you only made a mistake in the equation for the x-components. As you can see from the diagram you made, $F_{Nx}$ and $F_{Fx}$ are pointing to different directions so your equation for the x-component of the net force should be $-F_{Nx}+F_{Fx}=ma_c$, that way you'll get the right answer

[wraith dagger]

KonoEmllikDa

[cedar kiln]

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[wide forum]

hi, i'm here with a geometry / dynamical systems question that i've been looking to answer for a few weeks now but couldn't find any good material to study from.

the question is the following: why is the bucket handle continuum as defined geometrically on the plane (with semicircles ending on points of the cantor set, then reflected an shrunken, i'll add an image to illustrate) homeomorphic to the inverse limit of the full tent map on the interval?

[wide forum]

these are the first steps of the geometric construction on the plane

and this is the full tent map, it's just a piecewise linear map on the interval with slope 2 and -2

[wide forum]

this is an image with one more iteration, to illustrate

thanks in advance for any help or intuition, and i'd be happy to share more concrete details if needed

i've been scraping the whole written internet and i can't seem to find the explicit homeomorphism. i believe i understand the intuition, i understand the tent map does something like "kneading" but i'd like to be able to write the homeomorphism explicitly or at least read it somewhere in order to convince myself.

any hints are totally welcome

and thanks for reading

[cedar kiln]

@wide forum Has your question been resolved?

[cedar kiln]

@wide forum Has your question been resolved?

[cedar kiln]

@wide forum Has your question been resolved?

[cedar kiln]

@wide forum Has your question been resolved?

[cedar kiln]

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[still dune]

Pls help

[oak anchor]

Cauchy-Schwarz doesn’t work here?

another way would be to prove that $f(x) := {x^3 + b^3 \over 2} - ({x + b\over 2})^3 \geq 0$ for all x

[wraith dagger]

Médicis

[oak anchor]

by symmetry it would give a proof

[cedar kiln]

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[still dune]

Thanks guys

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[sturdy escarp]

how would you verify a multivariable limit exists with certainty?

[sturdy escarp]

testing paths just shows if limit DNE, no?

could spam a million paths and even if 999,999 of them equal the same thing one might still be off and it's gg

[dusk goblet]

yes

you’ll notice that you probably encounter questions that involve limits that don’t exist for that reason

it’s not as straightforward usually for showing it does exist but usually when it does it’s some simple continuous function

[sturdy escarp]

hmm

[dusk goblet]

squeeze theorem too

[sturdy escarp]

idk someone told me from a previous class year that they were asked to show the limit exists and show that a limit DNE even if all the paths give the same number

[sturdy escarp]

so just use eps delta def and squeeze theorem and cook smth up?

[silver rover]

🤦 🤦 thank you

[sturdy escarp]

I might get asked smth like this which is easy but if I needa come up with everything myself it might be gg

[dusk goblet]

this is asking for DNE

[dusk goblet]

sure but usually the eps delta would be really ugly

squeeze theorem is more common

[dusk goblet]

if all of the paths give the same thing then how would it not exist?

[sturdy escarp]

well I wouldn't have tested every path bc it's infinite, no?

surely one of them can mess up

someone in a video said that iirc

[dusk goblet]

well yea no finite number of paths will be sufficient

[sturdy escarp]

so just try random paths and if they're the same just conclude it exists?

[dusk goblet]

no

that’s not how math works

you cant conclude it exists because hey i tested a few and they all worked

at best it’ll be support for your guess

still doesn’t prove anything

[sturdy escarp]

okay makes sense

and ur saying to actually concretely prove it I'd needa do extra stuff like squeeze?

[dusk goblet]

usually yea

[sturdy escarp]

interesting

I might be given the graph as well

graph couldn't prove the limit exists right?

[dusk goblet]

proof by look at graph

[sturdy escarp]

😭 😭 😭

okay thanks knief

saving me once again

.close

[cedar kiln]

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[dusk goblet]

you’re welcome sir

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[sturdy escarp]

how can I solve stuff like this?

its rlly hard for me to visualize

[floral arrow]

Two of those don't use y, that should be easy to spot

Of the other three, two are symmetric (you can swap x and y and get the same function)

To differentiate between them, you can check one point, for example (0,0)

@sturdy escarp

[cedar kiln]

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[cedar kiln]

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[south tartan]

Hi, for this I had initally thought that for x = -5 the answer is a removable discontinuity since the factorised form is (x+5)(x-3) / (x+5)(x+5) so when its negative 5 both numerator and denominator will be 0, hence the removable discontinuity. But the answer I got from Khan Academy says its a vertical asymptote.

[south tartan]

I had seeked the help of AIs some say its vertical asymptote because after the cancellation of the (x+5) from numerator and denominator, hence (x-3)/(x+5) negative 5 only makes the denominator zero

but doesn't the simplified form has to match up with the original factorised form since they are the same if the original shows its a removable discontinuity then it is one.

[azure swift]

What is meant by this, I don’t follow

[south tartan]

from the original of (x+5)(x-3) / (x+5)(x+5) to the simplified form of (x-3)/(x+5)

[oblique flare]

Eh not really
You could use limits to make sure
The limit as x->-5 for both the original and simplified function does not approach the same value from the left and the right

[azure swift]

Are you aware of what is meant by a removable discontinuity?

It’s not that some factors cancel

The limit could still be undefined

[south tartan]

yes, from what i understand means both the numerator and denominator are both 0s so there isn't a spot on the x and y coordinate for a point

[azure swift]

If the limit at that point exists, then it’s a removable continuity

Maybe it helps if you graph it

[azure swift]

Well that doesn’t say much

Have you gone through the concept of continuity or limits?

[south tartan]

I'm learning the concept of rational functions on Khan Academy

it touched slightly on limit but mostly teaching the idea of discontinuties seperately

[azure swift]

Hm

[oblique flare]

Then try computing values close to x=-5 a little higher and lower and see how the function behaves

[azure swift]

Could it be that they are unaware that such behaviour is discontinuity

[south tartan]

sorry so regarding the question

its a vertical asymptote as after the simplified form of (x-3)/(x+5) the -5 makes it -8/0? hence its a vertical asymptote?

since there is no corresponding y value

[azure swift]

It might be helpful if you realise that it must be a vertical asymptote as the simplified form has something nonzero in the numerator and still zero in the denominator

[south tartan]

understood, so just to clarify cause i'm learning mainly through asking AIs concepts so it might have gotten wrong and messed my understanding

so in this case is it only the simplified form that takes precedent or does the original factorised form?

[azure swift]

I’d really recommend avoiding asking chat bots such questions, when it comes to learning concepts and so. More so to check problems if you really have to

[azure swift]

The both represent the same thing

As how 4/2 is the same as 2/1

But the latter is easier to deal with

Of course you loose some data

In that sometimes you lose the data that the rational function was say undefined at say some point x

But after simplification it now is defined for the same x

That means it’s a removable discontinuity

[south tartan]

but doesn't that mean the original of (x+5)(x-3) / (x+5)(x+5) of when x = -5 so (0)(-8)/(0)(0) so its essentially 0/0

so its a removable discontinuity?

[azure swift]

But 0/0 is intedermineed

[azure swift]

No that’s not a correct definition

In your case, how you’d realise (since you haven’t been taught limits yet)

Is that once you’ve simplified and the expression is now defined for say -5

Then that means it’s a removable discontinuity

But here we still have 0/0

And moreover after simplification you get something nonzero over 0

Hence -5 must be a vertical asymptote

Let me maybe give you some concrete examples to go through

Would that be okay?

[azure swift]

So yes in some sense the simplified form is what governs the behaviour

[south tartan]

oh its fine thanks, looking back on my notes i understand now

just came back from the military just messed up a bit of the definition

[azure swift]

Okay

[south tartan]

thanks!

[cedar kiln]

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[chrome drum]

https://cdn.discordapp.com/attachments/995774510164889783/1431303467997200504/image.png?ex=68fcecbb&is=68fb9b3b&hm=36676178fda3f94ff86017ae52d27ca0782de908b8e78bd6878b7961ae88f5cb& how can they do this

[chrome drum]

y / y^2 = 1/y

right?

[lone slate]

yes

[urban coral]

so what is wrong with this step, in your opinion?

[chrome drum]

shouldn't it be something like

((3x^2(y)(z^2)) / (9x^1(1/y)(z^3)))^-2

[chrome drum]

this?

[lone slate]

oh my god

[sacred anchor]

how did you even get this

[chrome drum]

[sacred anchor]

from y/y^2

[static phoenix]

how did you change y^2 to 1/y??

[urban coral]

...what, how did this happen

[lone slate]

holy math

[urban coral]

the math isn't mathing here. what were the steps that led to this?

actually, at this point, please show the original question

[urban coral]

I mean, what are you supposed to do with this? simplify?

[chrome drum]

yes

[static phoenix]

try writing y^2 as y * y and see what happens

[chrome drum]

i dont understand how this is possible specifically

if y / y^2 = 1/y

[urban coral]

this is correct

but you can think of it as $\frac{\text{some stuff}}{\text{some other stuff}} \cdot \frac{y}{y \cdot y}$

[chrome drum]

so why not have 1/y in the denominator and take out y in the numerator

[sacred anchor]

$\frac{y}{y^2} = \frac{\cancel{y}}{\cancel{y} \times y} = \frac 1y$

[static phoenix]

you're confused about how one of the y gets cancelled?

[wraith dagger]

Mika

[urban coral]

oops

there

[chrome drum]

confused on why it's left as y and not 1/y

is the 1/ implied or something

[urban coral]

if you insist on writing it as 1/y, you'll have to write it as its own fraction

[static phoenix]

see 1/(1/y) is the same as y

[sacred anchor]

and by extension, $\frac{ay}{by^2} = \frac ab \times \frac{y}{y^2} = \frac ab \times \frac 1y = \frac{a}{by}$

[urban coral]

$\frac{\text{some stuff}}{\text{some other stuff}} \cdot \frac{1}{y}$

[wraith dagger]

Mika

[urban coral]

like this

[sacred anchor]

for some two expressions a and b

[chrome drum]

but it's not 1/(1/y) it's 1/y

[static phoenix]

im trying to tell you that if you have (1/y) in the denominator which is what youre thinking then the y will go up in the numerator which is not the case according to the problem

[sacred anchor]

you can also think about it like $\frac{ay}{by^2} = \frac{ay \div y}{by^2 \div y} = \frac{a}{by}$

[static phoenix]

this is the answer to your problem

[chrome drum]

this feels wrong but i kind of understand now

thanks

.close

[cedar kiln]

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[cedar kiln]

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[snow solar]

why these two points are different, should their be the same polar coordinate?

[dusk finch]

it seems like it didnt interpret the coords as polar

[clever venture]

send the desmos graph

[dusk finch]

desmos thinks that those coords are cartesian

[snow solar]

just two points

[dusk finch]

by changing the polar graph setting, u only change the drawing in background

[clever venture]

desmos doesn't change the coordinates

yea that

[snow solar]

oh

[clever venture]

try putting r=(coordinate goes here)

wait no it doesn't nvm

[snow solar]

yea ;/

[dusk finch]

i dont think u can directly input polar stuff

but u can simply do
(rcos(theta), rsin(theta))

for points

for equations, you can do r = f(theta)

[snow solar]

ok, thanks

[clever venture]

may be a bit clunky but you can do this

[dusk finch]

yeah, shouldnt be cos(1)

[clever venture]

yea mb

[dusk finch]

should be 1cos(pi/4)

[snow solar]

yea not it works

i kinda wish it was simpler

but is fine

[clever venture]

do this actually

[snow solar]

ohh that works

[cedar kiln]

@snow solar Has your question been resolved?

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[worthy oasis]

Alguém me ajuda?

[worthy oasis]

Preciso de algum brasileiro

I need someone who is Brazilian, I'm studying for a math olympiad but I don't know anything

[flint cape]

[I'm not Brazilian nor do I speak Portuguese; but you gotta ask a more specific question than that]

[worthy oasis]

I want to learn first-degree equations and combinatorics, do you know how to use the translator?

[flint cape]

I mean if you're asking for a Portuguese speaker, I'm not going to pretend that I am, even with a translator

(also, idk, pick up a book or something?)

[worthy oasis]

Wdym with this?

[flint cape]

Like, surely you can Google for a textbook on those topics?

(also, please stop pinging me in response all the time)

[worthy oasis]

Sorry

Ok

Bye

.close

[cedar kiln]

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Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[autumn hamlet]

Hi

[autumn hamlet]

We've got
3^x+2 + 3^x+7 = (1/244)^-1

Need help with that

Oops

I forgot to tell u the question

[primal socket]

$3^{x+2} + 3^{x+7} = \left(\frac{1}{244}\right)^{-1}$

[wraith dagger]

USS-Enterprise

[primal socket]

This?

[autumn hamlet]

Yes

We need to find x

Which can be

And integret number

Like it can be positive or negative

[primal socket]

Yes

[autumn hamlet]

First thing I did

[primal socket]

Do you have any ideas

[autumn hamlet]

Is I flipped 1/244

To get 244^-1

Then I got stuck

[primal socket]

Well

[autumn hamlet]

On how to approach next

[primal socket]

That is incorrect

[autumn hamlet]

Oh

[primal socket]

What is $\left(\frac{a}{b}\right)^{-1}$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

a^-1/b^-1

[primal socket]

Well, sure but

You said you flipped it right

[autumn hamlet]

Yes

Since it's 1/244

[primal socket]

Yes

And after 'flipping' what do we get

[autumn hamlet]

Oh

[graceful karma]

1/244 to the power of -1 = 244

[primal socket]

Thanks, but I believe HH2 is trying to figure this out on his own

Anyway, HH2

[autumn hamlet]

But isn't 1 odd number

[primal socket]

Generally, when we raise something to the power of -1

[autumn hamlet]

So why would it give 244

[primal socket]

We are taking the reciprocal value of that number

[autumn hamlet]

Oh wait

Sorry mb

The power is negative

[primal socket]

Do you know what a reciprocal is

[autumn hamlet]

Not the number itself

[autumn hamlet]

Nope cause don't know English math terms

I only know factorise

[primal socket]

Right

The reciprocal of a number is 1 divided by that same number

For example, the reciprocal of 5 is 1/5

[autumn hamlet]

Oh ok

Got it

Now I know what it means

[primal socket]

And this can be written as 5^(-1)

Which is 1/5

[graceful karma]

surely you'd split the exponents of the top

[primal socket]

So when we raise something to the power of -1 we are taking the reciprocal of that number

[autumn hamlet]

Ok

[primal socket]

And in the case of fractions it ends up to just "flipping" the fraction

Because, if we have

$\left(\frac{a}{b}\right)^{-1}$

[wraith dagger]

USS-Enterprise

[primal socket]

As we said, we take 1 and divide it by this number

So we get

$\frac{1}{\frac{a}{b}}$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

Oh ok

[primal socket]

And this simplifies to $\frac{b}{a}$

[wraith dagger]

USS-Enterprise

[primal socket]

Right?

[autumn hamlet]

Yes...?!$$$

[primal socket]

So $\left(\frac{1}{244}\right)^{-1}$

[wraith dagger]

USS-Enterprise

[primal socket]

Is actually 244

So now we have

[autumn hamlet]

Ok

[primal socket]

$3^{x+2} + 3^{x+7} = 244$

[wraith dagger]

USS-Enterprise

[primal socket]

How do we continue

[autumn hamlet]

Hmm

Can we

Break down the three

So its

3^x . 3² + 3^x . 3⁷?

[primal socket]

Yes, exactly!

Then?

$3^x \cdot 3^2 + 3^x \cdot 3^7 = 244$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

We do this:
3^x (3²+3⁷)

?

[primal socket]

Yes, we factor out 3^x

$3^x \cdot \left(3^2 + 3^7\right) = 244$

[wraith dagger]

USS-Enterprise

[primal socket]

Then?

[autumn hamlet]

Now we calcul?

[primal socket]

Lead the way

[autumn hamlet]

3² is 9 and 3⁷ is 2187

Is that correct?

Or

Should I not have calculated

[primal socket]

Well

Can you see how we can continue if we calculate them

[autumn hamlet]

We get 2196

Then my brain gets stuck

:3

Any help? :3

[primal socket]

Or rather, if we can?

Sorry, I am on a train and my mobile data cuts out here and there

[autumn hamlet]

Oh

[primal socket]

Yes

[autumn hamlet]

Lol

[primal socket]

So we have

[autumn hamlet]

It's ok

[primal socket]

$3^x \cdot 2196 = 244$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

Yes

[primal socket]

No ideas?

[autumn hamlet]

We can

Break down 2196?

[primal socket]

Hmm

But we just worked so hard to get it into one number

[autumn hamlet]

Hmmm

[primal socket]

Factoring out 3^x, adding, etc.

[autumn hamlet]

Right

[primal socket]

Why would we undo the work

Let's do this

[autumn hamlet]

Should we

Make 2196 on the other side

[primal socket]

How?

[autumn hamlet]

Make 2196 minus

So -2196

[primal socket]

Subtract 2196 on both sides?

[autumn hamlet]

Yes

[primal socket]

Let's try

We get

$\left(3^x \cdot 2196\right) - 2196 = 244 - 2196$

[wraith dagger]

USS-Enterprise

[primal socket]

Right?

That's subtracting 2196 on both sides

[autumn hamlet]

Oh uhh I

When we usually do it

Or

Uhm

[primal socket]

Don't worry

This is a common mistake

I know you want to get rid of the 2196 by subtracting it

But the thing is

[autumn hamlet]

🤖 🧠 🛑

[primal socket]

Fhe 2196 is first being multiplied by 3^x because of the order operations

[autumn hamlet]

Yes that is what got me

[primal socket]

And we can't easily subtract numbers from an unknown product

It's like what is (a - 20)

We have no clue because what is a?

The same thing is here

a is 3^x

And 20 is 2196

[autumn hamlet]

x

[primal socket]

So

[autumn hamlet]

Yes

[primal socket]

For the sake of simplicity

Let's say that some variable b equals 3^x

We get $b \cdot 2196 = 244$

[wraith dagger]

USS-Enterprise

[primal socket]

This changes nothing because we said b = 3^x

Do you see this is the same thing?

It's just b instead of 3^x

[autumn hamlet]

Hmm

That's right hopefully and I'm not getting tricked by you

Go on :3

[primal socket]

I am just trying to show you how to solve equations like these in general

You won't ever do what we are doing here

[autumn hamlet]

Yes ik

[primal socket]

Just trying to show you why we do

But, please answer my question

You understand the two equations are the same thing right

[autumn hamlet]

Technically

Yes

[primal socket]

Okay

It doesn't matter it's b

Jt could have been y

Or 16000c

It doesn't matter

Anyway

$b \cdot 2196 = 244$

[wraith dagger]

USS-Enterprise

[primal socket]

This is a variable multiplied by a number

[autumn hamlet]

I see something

2196b?

[primal socket]

Yes!

We have 2196 of b

[autumn hamlet]

2196b/244

[primal socket]

Or 2196 of 3^x

Exactly!

We have 2196 times b, or 2196 times 3^x

[autumn hamlet]

9!!!

[primal socket]

So we can very rightfully divide both sides of the equation by 2196

So we get b, or 3^x alone

[autumn hamlet]

9..

[primal socket]

Hang on

My mistake

[primal socket]

What did you do here

We have $2196b = 244$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

Divided 👀

[primal socket]

What is b?

[autumn hamlet]

3^x

[primal socket]

Okay

But if we solve for b

You said we divide

[autumn hamlet]

Yes

[primal socket]

We divide both sides by 2196

To get b alone

Right?

[autumn hamlet]

Oh right

My mistake

I got hyped

Lol

[primal socket]

😅

[autumn hamlet]

Yes so we get b=244/2196

[primal socket]

So we end up with $b = \frac{244}{2196}$

[wraith dagger]

USS-Enterprise

[primal socket]

Exactly

Can we simply the fraction?

[autumn hamlet]

Sure

Let me get

The best divider for both

9

[primal socket]

Hmm

[autumn hamlet]

Wait

[primal socket]

What is 244 divided by 9

[autumn hamlet]

No no

Calculator did my dirty

[primal socket]

😂

[autumn hamlet]

Wth

[primal socket]

<@&268886789983436800>

[autumn hamlet]

Scammer

Les go

I'ma get scammed today

[primal socket]

Let's get back

[autumn hamlet]

I got 4

[primal socket]

Simplify 244/2196

[autumn hamlet]

Since 244/4 is 61

And

2196/4 is 549

[primal socket]

Yes

But we can go further

61/549

[autumn hamlet]

Both

Divided by 61

1/9

[primal socket]

Yes

Exactly

So

$b = \frac{1}{9}$

[wraith dagger]

USS-Enterprise

[primal socket]

Now we said b = 3^x

So we can pop it back in

[autumn hamlet]

Yes

[primal socket]

$3^x = \frac{1}{9}$

[wraith dagger]

USS-Enterprise

[primal socket]

Now what?

[autumn hamlet]

Well

We can do

That reprocial

Or idk

Reprocecal

Ugh

[primal socket]

Yes, exactly

[autumn hamlet]

Anyways

[primal socket]

Exactly why I explained it the long way before

What is 1/9

[autumn hamlet]

9-¹

[primal socket]

Yes

$3^x = 9^{-1}$

[wraith dagger]

USS-Enterprise

[primal socket]

We have an exponent

3^x

And an exponent

9^(-1)

When are two exponents equal

[autumn hamlet]

3²x-1?

[primal socket]

What?

[autumn hamlet]

So it's 3^-2?

Uhh

I simplified 9

[primal socket]

Ah

Yes

We have $9^{-1}$

[wraith dagger]

USS-Enterprise

[primal socket]

We (hopefully) also know 9 = 3^2

In which case

[autumn hamlet]

(3²)-¹

[primal socket]

Yes

What does the power of a power rule say

[autumn hamlet]

We multiply the powers??

[primal socket]

Exactly

[autumn hamlet]

3-²

[primal socket]

$\left(x^{a}\right)^{b} = x^{a \cdot b}$

[wraith dagger]

USS-Enterprise

[primal socket]

So

$9^{-1} = 3^{-2}$

[wraith dagger]

USS-Enterprise

[primal socket]

And we get

$3^{x} = 3^{-2}$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

Yup

[primal socket]

When are two exponents equal?

[autumn hamlet]

is that a question

Uhm

[primal socket]

Yes

[autumn hamlet]

We add em powers

?m

[primal socket]

Say we have

$a^{x} = b^{y}$

[wraith dagger]

USS-Enterprise

[primal socket]

When will these exponents be equal

Exponents being the whole number (the base and the power)

[autumn hamlet]

Oh

Ye I wish they had tought us this instead of natural and integret numbers

[primal socket]

Everything's important

Let's try it this way

[autumn hamlet]

Ye but how will I know if I never saw it on class : 😭

[primal socket]

Is $3^2$ equal to $3^5$?

[wraith dagger]

USS-Enterprise

[autumn hamlet]

No ofc

They need to have same power

[primal socket]

How about $2^5$ and $4^5$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

Since they are the same base

[autumn hamlet]

They have same power

[primal socket]

Yes

[autumn hamlet]

But not same base

[primal socket]

So

If we combine the two

In general

When are exponents equal

[autumn hamlet]

When they have same base and power

[primal socket]

Exactly

In the first one we needed the same power

In the second the same base

[autumn hamlet]

Epikness runs in ur blood ngl

[primal socket]

So

$a^{x} = b^{y}$

[wraith dagger]

USS-Enterprise

[autumn hamlet]

5 minutes = whole lesson

[primal socket]

This is true when a = b and x = y

In our case

[autumn hamlet]

Yes

[primal socket]

$3^x = 3^{-2}$

[wraith dagger]

USS-Enterprise

[primal socket]

We already have a = b, 3 = 3

[autumn hamlet]

a=b

Yes

[primal socket]

So we just need x = y

Or $x = -2$

[wraith dagger]

USS-Enterprise

[primal socket]

And there's your solution

[autumn hamlet]

Les go

[primal socket]

🙂

So to sum up

With these equations

[autumn hamlet]

Tysm

[primal socket]

You will usually just be simplifying stuff at first

Then dividing

And then making sure the two sides of the equations have the same base

And then comparing the powers

That's usually as far as you will go in school

[autumn hamlet]

Oku

[primal socket]

But I just wanted to make every step clear

Why we do it

Because I believe if you understand why, it makes it insanely easier to do it in general

[autumn hamlet]

You should be a professor at a school (maybe u are one 🤔 )

[primal socket]

Well I am in first year university right now

We'll see

😄

[autumn hamlet]

Tell em to recruit

:3

[primal socket]

But thanks for the kind words

Have a good day

[autumn hamlet]

Np

[primal socket]

See you around

[autumn hamlet]

Cya

.close

[cedar kiln]

Channel closed

Closed by @autumn hamlet

Use .reopen if this was a mistake.

[cedar kiln]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[cinder brook]

hello

[cinder brook]

The longer spatial diagonal of a regular hexagonal prism is inclined to the plane of the base at an angle of 45°. Find the surface area and volume of the prism if the area of ​​its base is 12√3 cm².

[cedar kiln]

@cinder brook Has your question been resolved?

[hollow trail]

what have you tried so far?

[cinder brook]

is this correct?

[cinder brook]

<@&286206848099549185>

[carmine bronze]

Your drawing looks accurate.

[cinder brook]

i placed the 45° given in the question accurately, right?

so as for volume, i found 48sqrt6 cm^3

and as for area, 24sqrt3 + 96 cm^2

i don't have answer key

[carmine bronze]

I don't think there will be a specific numerical answer.

Because no height is given.

[cinder brook]

?

if i placed it correctly

then both of those triangle angles are 45°

and therefore both are 4sqrt2

which means height is 4sqrt2

[carmine bronze]

Imagine the catty corner of the hexagon can be at any point on the diagonal line you made.

[cinder brook]

i'm not sure i fully understand

are you saying the diagonal i drew is wrong?

[carmine bronze]

No, the diagonal is correct. I'm saying that the top hexagon can slide to any point on the diagonal line and be at a 45 degree angle from the bottom catty corner.

[cinder brook]

so what's the answer if it's not numerical?

did you find it?

[carmine bronze]

It will contain a variable that represents the height of the prism.

[cinder brook]

the 2sqrt2 is already given

that way it would be way too easy

[carmine bronze]

If you give me a moment or several moments, I will make a Geogebra file that demonstrate what I mean.

[cinder brook]

if the answer is just volume=base area.h

[cedar kiln]

@cinder brook Has your question been resolved?

[cinder brook]

<@&286206848099549185>

[cinder brook]

okay, new question

no one's helping with this one anyway

i need to find the volume of the truncated part

the one with 12

the one with 8 is just my extension

<@&286206848099549185>

[carmine bronze]

My apologies, I got sidetracked. Anywhos, here is a Geogebra file that shows the issue I mentioned.
https://www.geogebra.org/classic/hmdwmzy2

[cinder brook]

no worries, and thank you for it

[cinder brook]

can you check the new question if you have time?

[carmine bronze]

So you have a pyramid that is truncated with side lengths of 12 and 8 units?

And the sides of the base are 10 units long?

[cinder brook]

this is it

[carmine bronze]

Have you tried Googling the formula for the volume of a truncated triangular pyramid?

[cinder brook]

i did

and i solved it that way

the problem is

i get different answers when i use that formula and when i extend the pyramid and calculate whole extended pyramid then remove the extended part

the answers have to match up

and i double-checked, more like hundred times checked all my calculations already

[carmine bronze]

Do you recalculate the heights?

[cinder brook]

i used correct height for both solving methods

so it's not that

the formula method gave me 78sqrt11

and the extension method gave me 4750/81 . sqrt11

4750/81 is like 58, so 58 and 78

it's not an astronomical number that's what i mean

<@&286206848099549185>

[carmine bronze]

What do you mean by extension?

The part that wsa cut off?

[cinder brook]

extending the pyramid into a full pyramid

finding volume of full pyramid

and then removing the extended small part to find the part we want

[carmine bronze]

Can you show your work?

[cinder brook]

finding the volume of whole pyramid and then multiplying by 19/27

as for finding whole volume

base area is 10^2.sqrt3 /4 = 25sqrt3

height is sqrt1100 / sqrt3

i didn't forget 1/3 in the formula

solving from this way gives us 4750. sqrt11 / 81

solving it from the other way(formula for truncated triangular pyramid) on the other hand

gives us 78.sqrt11

[carmine bronze]

Give me a moment or two to work the problem out myself.

Ahhh, my internet is so slow right now. 😠

[cinder brook]

it's okay, take your time

i can ping helpers right

[carmine bronze]

Yes.

[cinder brook]

more than 15 more than 35 minutes actually

<@&286206848099549185>

[carmine bronze]

What did you get for the truncated area?

Your height does not seem correct.

[cinder brook]

which one?

[cinder brook]

this?

that √1100/√3 is the height of whole extended pyramid

i used that one for calculating the whole pyramid and then multiplied by 19/27(which is removing the small part)

the height for truncated area is 2√33

so, height of truncated pyramid at the bottom is 2√33
the height of extended pyramid is √1100/√3

[cedar kiln]

@cinder brook Has your question been resolved?

[cedar kiln]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

[cedar kiln]

Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[grave tulip]

doing question 3 right now and i've figured everything out except for how to get the x outside the absolute values in the hint because in part a it is inside

[cedar kiln]

@grave tulip Has your question been resolved?

[royal finch]

It's not clear to me what you have done in part (c) that is causing you problems just from that description.

[grave tulip]

i think im good now ty

.close

[cedar kiln]

Channel closed

Closed by @grave tulip

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Remember:
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• Show your work, and if possible, explain where you are stuck.
• Do not immediately ping people or roles. After 15 minutes, feel free to ping <@&286206848099549185> once.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[north epoch]

I am trying to project screen coordinates onto an isometric plane

but.... for some reason the math gets less accurate the further away from center it gets

[crimson sedge]

What does isometric mean

[north epoch]

https://www.youtube.com/watch?v=04oQ2jOUjkU

Ummm

instead of a square grid, everything is kinda smooshed

at this point i just copied this guys code, because mine didn't work

but even when i copied his code it didn't work

const i_x = 1.0;
const i_y = 0.5;
const j_x = -1.0;
const j_y = 0.5;

const w = 32.0;
const h = 32.0;

const Matrix = struct {
    a: f32,
    b: f32,
    c: f32,
    d: f32,
};

fn invert(a: f32, b: f32, c: f32, d: f32) Matrix {
    const det = 1.0 / (a * d - b * c);
    return .{ .a = det * d, .b = det * b * -1.0, .c = det * c * -1.0, .d = det * a };
}

pub fn tile_to_screen(tile: rl.Vector2) rl.Vector2 {
    return rl.Vector2{
        .x = tile.x * i_x * 0.5 * w + tile.y * j_x * 0.5 * w,
        .y = tile.x * i_y * 0.5 * h + tile.y * j_y * 0.5 * h,
    };
}

pub fn screen_to_tile(screen: rl.Vector2) rl.Vector2 {
    const zoom = 1.0 / eng.use_camera().camera.zoom;
    const a = i_x * 0.5 * w;
    const b = j_x * 0.5 * w;
    const c = i_y * 0.5 * h;
    const d = j_y * 0.5 * h;
    const m = invert(a, b, c, d);

    const screen_x = screen.x * zoom;
    const screen_y = screen.y * zoom;

    return rl.Vector2{
        .x = screen_x * m.a + screen_y * m.b,
        .y = screen_x * m.c + screen_y * m.d,
    };
}

feel free to close if too much code lol

[royal finch]

It's pretty hard to tell without spending a bunch of time printing statements, but assuming the linear algebra is correct it could be anything from floating point precision to numerical instability on calculating the determinant.

I would probably start by at least printing out the largest and smallest eigenvalues of the matrix you are inverting where the errors are starting to appear and making sure they aren't too weird

The calculations are pretty simple, the only thing is whether the math itself is wrong, or if that division step is problematic

[north epoch]

of the determinate?

[royal finch]

yes

Or if you don't want to spit out a bunch of eigenvalues, just replace that step entirely with a numerically stable algorithm from a library and make sure that doesn't solve the issue immediately.

[cedar kiln]

@north epoch Has your question been resolved?

[cedar kiln]

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[wild plaza]

how do i do arctan -> infinity?

[pastel vault]

you already found the integral when the lower limit is t

so yeah, taking the limit as $t \to -\infty$, you do need to find the value of $\arctan(+ \infty)$

[wraith dagger]

south

[pastel vault]

finding y when x = +infinity
is the same as finding which value of x gives tan(x) = +infinity

[cedar kiln]

@wild plaza Has your question been resolved?

[wild plaza]

oh i forgot how the arctan graph looked like, it makes more sense now

[cedar kiln]

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[pastel vault]

no worries!

you can figure out the graph from properties of inverse functions is what I mean

[cedar kiln]

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[misty comet]

I was recently playing around with Sigma notation until i decided to graph it on desmos using variables, I noticed that when the variable was to the right, it graphed perfectly fine but when put above the notation, it made this weird pattern, the number to the right determined the distance from each line but I could not understand what the width meant? Why is the width in-between decimals like 5.5 and 6.5? And is there anymore information on this topic

[bold lotus]

do you know what sigma notation is?

[tropic oxide]

do you know how sigma notation works regardless of its specific/edge-case behavior in desmos

[misty comet]

No I learned it from a friend who told me you can use sigma notation to multiply two numbers together overcomplicatingly

[bold lotus]

that's not correct

[tropic oxide]

yeah it's a bit... no

[bold lotus]

its just a way of expressing a sum

[tropic oxide]

sigma notation at its core is a way of writing down sums (i.e. additions of lists of numbers)

especially long ones, especially ones where the terms follow a pattern

[bold lotus]

have you ever seen something like this: S = 1 + 2 + 3 + ... + (n - 1) + n

[tropic oxide]

have you done any computer programming? @misty comet

[misty comet]

Yes

[misty comet]

Nope

[tropic oxide]

aight nevermind that then

[bold lotus]

that's one way to express a sum of adding up 1 to n right

[misty comet]

I did this in Algerba I i think

[tropic oxide]

right so

https://www.youtube.com/watch?v=5jwXThH6fg4

[bold lotus]

but its really verbose so sigma notation was made

[tropic oxide]

perhaps read this

[tropic oxide]

the verbosity really starts to show if you try to talk about a sum whose terms have a more complicated pattern than like, geometric series

[misty comet]

So its just a simplified version of that formula?

[bold lotus]

that's not a formula

its a way to represent the sum without writing out all the values

we show the first and the last terms and write ... for the middle

I recommend you watch the video

[misty comet]

Okay I'll watch the video especially because i see the symbol everywhere

Now that I'm getting more into math

.close

[cedar kiln]

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[cedar kiln]

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[hexed vortex]

when they say verify does that mean find every possible form of G and show each one has a element of order 180

[tropic oxide]

can you show us the corollary itself

[tropic oxide]

ok then you need to show that any ab group of order 1080 has a subgroup of order 180

prime factorizations of each one would help

[hexed vortex]

yeah so I would have to show every form of G has an element of order 180?

I got 2^3x3^3x5 for. the prime factorization

so now I need to find all isomorphism classes and show each has a element of order 180