help-0

[rotund junco]

.close

[lone heart]

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[alpine sable]

What is v sub 0

[rotund junco]

initial velocity

[gilded magnet]

Wdym for a point o

From a point O?

@rotund junco ?

[rotund junco]

yh

[alpine sable]

O meaning ground level ig

[rotund junco]

i had this question in my exam today and i had a heart attack

[gilded magnet]

Isn't √3Vo as a whole initial vel

[rotund junco]

i could only get part 1

[alpine sable]

I'm having a heart attack just reading the equation

[rotund junco]

yeah

not only that

i had like 15mins left and i read it

[alpine sable]

I'm in alg 1 so this confuses me a lot

[rotund junco]

idk what alg 1

[gilded magnet]

Well it's just integration differential with variable accN

[rotund junco]

yeah i remember than

that

[alpine sable]

Basically basics of algebra

[rotund junco]

idk if youre in australia

[gilded magnet]

U might wanna put a=vdv/dx or smthg

[rotund junco]

but this is from the highest level of school math

yeah i did that

for pt 1

but for pt 2 i was frozen

[gilded magnet]

I just got over with hs...we have harder problems in jee advanced

[rotund junco]

i was just wondering if anybody could help

[gilded magnet]

India

[rotund junco]

whats hs?

[gilded magnet]

High school

[rotund junco]

lucky

[alpine sable]

💀

[rotund junco]

looooooooool

[alpine sable]

Fr

[rotund junco]

what would you do for pt 2?

any ideas

[gilded magnet]

Ok

[rotund junco]

maybe itll make me see a different perspective

[gilded magnet]

So mg down air resis up?

[rotund junco]

yeah

no

[gilded magnet]

While moving from topmost to down

[rotund junco]

yeah yeah

but for part 1 theyre both down

[gilded magnet]

Starting with hMax

[gilded magnet]

I thought u only wanted help with part2

[rotund junco]

h max is now zero

yeah part 2

mbmbmbmb

im fried

[lone heart]

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[honest halo]

can you get log(b) / log(c) = a from log_a(b) = c?

[echo socket]

You mean log(b)/log(a) = c?

[thorny patio]

You get

a^c = b

from log_a(b) = c

[honest halo]

oh ok thankd

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[hybrid meadow]

for a set (R/{0} x R) does it have a boundary or boundary point?

[hybrid meadow]

<@&286206848099549185>

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[hybrid meadow]

<@&286206848099549185>

[tardy stag]

Do you mean R \ {0} like R without 0?

R / {0} makes me think of a quotient space which I don't think is what you mean but I'm not sure

[lone heart]

@hybrid meadow Has your question been resolved?

[hybrid meadow]

yup

[tardy stag]

what definition do you know for set boundary?

[hybrid meadow]

basically that it includes points from the set and its complement

also not empty set

[tardy stag]

mmk, we need a universal set then (so we can talk about boundary)

[tardy stag]

hmm that doesn't seem right, nothing is in a set and its complement that's the definition of complement

should have something to do with limit points maybe?

what do you think the boundary of this set should be?

[hybrid meadow]

im not quite sure what you mean by that

[tardy stag]

there's like three definitions of boundary so I'm not sure which one you're using

[hybrid meadow]

i was thinking 0 since every point beside it is in the set while it isnt

[hybrid meadow]

clos (R/{0})

[tardy stag]

well, the points in this set are ordered pairs

[hybrid meadow]

yup so basically all points with x as 0

[tardy stag]

yeah that's what I think as well

can you prove that every other point is not part of the boundary?

like if I gave you (x,y) where x != 0, how can you show that that's not part of the boundary of S

[hybrid meadow]

yup because if take because it would be of the form (R,R) meaning that its fully in the set

and all surrounding points are too

[tardy stag]

what do you mean by "all surrounding points are" exactly?

[hybrid meadow]

https://www.statisticshowto.com/boundary-point-definition-examples/

like this

[tardy stag]

(I'm assuming this is for topology?)

[hybrid meadow]

yeah, not very deep though in terms of coursework

[tardy stag]

still, you should have a workable definition of what it means for a point to be in the interior of a set

[hybrid meadow]

just this really

[tardy stag]

nothing about an open ball around the point being a subset of S?

that's how we usually define interior

[hybrid meadow]

i mean ive visualized it like a disk surrounding any interior points and as long you can move the disk with it still being in the set its an interior point

but similar to that yeah

[tardy stag]

that works for the real line/plane/space yeah

but if you're in topology you'll soon get some cool counterintuitive results :D

[hybrid meadow]

ahh ok, although im not sure how deep we'll get into that. Anyways what would then the boundary be though, just (0,y)?

[tardy stag]

yep! {(0,y) | y \in \R}

[hybrid meadow]

alright thanks

/close

[tardy stag]

?close

[hybrid meadow]

?close

$close

[surreal meadow]

you have to ask nicely

[hybrid meadow]

.close

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[hybrid meadow]

🫡

[lone heart]

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[vernal arch]

How to do this?

[twilit dune]

whats the task

[vernal arch]

solve it

[tacit arch]

<@&268886789983436800> troll

[vernal arch]

Im not a trool

troll*

i swear

I came here for help

[surreal meadow]

do what

find nontrivial zeroes perhaps?

[thorn tapir]

uhh

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[wintry kite]

Let's consider the body K = {(x, y, z) ∈ R^3: x^2 + y^2 ≤ z^2, 0 ≤ z ≤ 1}. The density of K is given by the function %(x, y, z) = x^2 + y^2 + z^2.
We need to find the moment of inertia of the body about the z-axis. can someone help pls idk what to do

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@wintry kite Has your question been resolved?

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[hushed wolf]

sin15, x, cos15 are the first three terms of a geometric sequence that has all positive terms x=?

[lone heart]

Show your work, and if possible, explain where you are stuck.

[hushed wolf]

i tried something like r²sin15=cos15
r²=cot15
r=√cot15
x=sin15√cot15
x=√cos15sin15=√2/2√2sin15cos15
=√2/2*√sin30=√2/2*√2/2=1/2

[echo socket]

It would have been faster if you used the fact that a term of a geometric sequence squared is equal to the product of its neighbouring terms

[mellow grail]

what..? (the last step)

[echo socket]

So, x^2 = sin(15) * cos(15) in this case

[hushed wolf]

it is the same thing no?

yeah it would be faster

[hushed wolf]

2sinacosa=sin2a

[mellow grail]

Oh

[hushed wolf]

so x=1/2 right?

[mellow grail]

yeah ig

works out

2nd one is sin 15/x, first is x/cos 15

they are equal

[hushed wolf]

okay thanks

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[plain nexus]

If we have an N x N matrix A

[plain nexus]

and we want to change it's base

we will write it as B^-1 * A * B

why is it written like that?

[plain flame]

B is your change of base matrix

so suppose B is your change of base matrix from D to E

then you would write B_ED

and (B_ED)^(-1) = B_DE

[plain nexus]

oh

[plain flame]

so suppose you have a vector in base D

[sonic harbor]

In picture

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@plain nexus Has your question been resolved?

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[boreal sigil]

What is 3×(-4)+2

[analog falcon]

good question

what do × and - and + mean?

[boreal sigil]

x is times

• minus

[analog falcon]

yes

[boreal sigil]

or what do u mena

my english sucks

[analog falcon]

yeah I mean just that

so if you know the operators

then can't you find the answer?

[boreal sigil]

i get it to -10

but i cant tell if thats correct

[analog falcon]

there is this thing called BODMAS that defines the order of what operators you do first

see the last 4 alphabets
they stand for
division | multiplication | addition | subtraction

so first you'll calculate 3×(-4)

and then add 2 to that

[boreal sigil]

10 right

[analog falcon]

nope

[boreal sigil]

its not -10?

[analog falcon]

it is -10

[boreal sigil]

Ok thanks

Really suck at math

[analog falcon]

We've all been there 🙂

[boreal sigil]

Lol

thanks tho take care

i just needed confirmation

[analog falcon]

Welcome

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[ivory fern]

Let A and B be two 100×100 real matrices such that A^2 + B^2 + AB = 0. Show that the determinant of AB - BA equals zero

[sonic harbor]

Show work 😄

[ivory fern]

I don't have it, this is a question of an exam I did yesterday and I didn't keep what I tried

But I basically just tried to multiply the equation of the hypothesis with A and B and tried to add things

Also tried to see if I could do something related to eigenvectors but I had no luck

[sonic harbor]

I've also just done my exams in linear algebra. Did pretty well.

I can't see the answer, but we can try to work the problem together? Or I can let the chat go to an expert if any show up

$$ A^2 + B^2 + AB = 0$$
Should give us:
$$ (A + B)^2 = 0 $$ right? This should be useful

[ocean seal]

zeimon4

[vale wigeon]

no

[marsh fog]

AB

[vale wigeon]

this is not even true if A and B commute

[sonic harbor]

The expert came

[vale wigeon]

which we do not know if they do

[ivory fern]

No, (A+B)^2 = BA

[marsh fog]

yh mb BA

[sonic harbor]

Cool, worked mostly with eigenvalues / vectors and determinants along with complex numbers. I'll just follow the chat 👀

[ivory fern]

For example it would be sufficient to prove that A and B have a common eigenvector

But that doesn't look useful

[lone heart]

@ivory fern Has your question been resolved?

[ivory fern]

<@&286206848099549185>

[marsh fog]

I'm still here lol and will happily admit I'm pretty much stuck

This is a gross question

[ivory fern]

Yes
I tried it for more than an hour

At this point I believe the question might not be correctly written and perhaps there is a counterexample

[outer elm]

u guys solved it?

[marsh fog]

Nope.

[outer elm]

okay lemme try

why does it matter that theyre 100x100

or real

[marsh fog]

I suspect 100x100 is just to prevent direct computation, although I can't rule out it being relevant entirely. Real doesn't really matter I don't think, could be over C.

[outer elm]

i see

[ivory fern]

The 100 might be inportant at some point, perhaps there are roots of unity or something and you need 100 to be congruent to some number modulo some other number

But maybe not

[outer elm]

hmmm seems unlikely but perhaps

[marsh fog]

I don't see how you could get that kind of information from this setup.

[ivory fern]

And it might be important that they are real at some point for some reason but maybe not

[ivory fern]

It is unlikely, but it would not be the first time I see roots of unity appearing in linear algebra

[upbeat hornet]

did you remember the identity wrong?

[outer elm]

i rlly hope so lol

im losing my mind over this

[marsh fog]

The closest thing I found online was the following olympiad problem: https://math.stackexchange.com/questions/4337421/is-it-necessary-that-detab-ba-0

This problem is even more difficult, in my opinion, since the form $A^2 + B^2 -2AB = 0$ is much more workable.

[ocean seal]

kimbo7

[marsh fog]

So it seems possible to me that it was either misremembered or there is a crazy trick.

I should add that the method in the olympiad problem sort of gets you close to something meaningful, perhaps contradicting invertibility does work but in a different way, but I've tapped out for now.

[lone heart]

@ivory fern Has your question been resolved?

[ivory fern]

No, I did not misremember it. I am completely sure the question said A^2 + B^2 + AB = 0

[marsh fog]

Fair enough! No idea then, this is a tough question.

[ivory fern]

But again, the exam might have been wrongly written

[ivory fern]

And can't this be adapted to the case A^2 + B^2 + AB = 0?

[sonic harbor]

Currently on my 26th awake hour. Idk how I could say such thing as the above, yet I did.

[marsh fog]

By all means try!

[lone heart]

@ivory fern Has your question been resolved?

[outer elm]

im forwarding this question to my uni friends

dont fret fellas

[flat dawn]

interesting question

another way to write it is

if AB = -(A^2+B^2) then det(AB-BA) = 0

may be useful as you have AB on the right side

you can show that

A(AB-BA) = A^2*B -ABA = ... = B^2*A - AB^2

similary you can show that

(AB-BA)B = BA^2 - AB^2

[flat dawn]

can we show this in a limited case, i.e. det(A) = 0?

[ivory fern]

You can separate cases det A = 0 and det A ≠ 0, but at the end both have to be proved

[flat dawn]

can we do at least det(A) = 0?

it would imply det(A^2 + B^2) = 0

[ivory fern]

Why?

Ah nevermind I see why

[ivory fern]

We have to prove it

[flat dawn]

just trying to see if there's an easier case from which we can learn

alternatively would be interesting to show det(AB-BA) >=0 or <=0

[ivory fern]

This might be useful too

[worn fox]

The fact they specifically want you to show det(AB-BA) = 0 leads me to think we shouldn't just be trying to show they commute

Otherwise why not just ask to show AB = BA

[ivory fern]

I agree

[worn fox]

My only other thoughts are somehow showing AB - BA has 0 as an eigenvalue

[flat dawn]

indeed

[worn fox]

Maybe with some char/min poly shenanigans but nothing looked nice when I tried

[flat dawn]

anything interesting that can be said about A^2 + B^2?

honestly I wouldn't be surprised if the statement is false either

[ivory fern]

Neither I
Maybe we could look for counterexamples for A, B with 2, 3 or 4 columns and see if they can be adapted to 100×100

[flat dawn]

@ivory fern if you don't mind, ping me when you have the solution as I'm curious myself

[ivory fern]

Ok, I'll tell you if I solve it

[lone heart]

@ivory fern Has your question been resolved?

[lone heart]

@ivory fern Has your question been resolved?

[ivory fern]

I proved that A^3 = B^3, perhaps that is useful

[worn fox]

how did you do that

@ivory fern

[ivory fern]

AB = -A² - B²
A²B = -A³ - AB²
A²B = -A³ + (A² + B²)B
A²B = -A³ + A²B + B³
0 = -A³ + B³

[vapid shuttle]

the factorization in your third line is incorrect

[ivory fern]

Why?

I just substituted -AB with A² + B², which is the first line

[vapid shuttle]

got it didn't realize that

[alpine sable]

usually for these kinds of problems i like to pretend that the matrices are all like, real numbers, or indeterminates, or whatever. so i would factor A^2 + B^2 + AB to get (A - zB)(A - (1/z)B) for z = -1/2 + i sqrt(3)/2

so looking at those factors might be helpful

(obviously this doesn't apply word for word. A^2 + B^2 + AB isn't actually that. but hey)

like my favorite example of how i think about matrix problems is the following classic

if A is nilpotent then I - A is invertible

well okay so like, you want basically 1/(1 - x) with A in place of x

we know that that's equal to 1 + x + x^2 + ...

if A is put in for x then nilpotency kills all but finitely many terms. so this works

i try to throw this at literally everything

[brisk thunder]

there is indeed a relevant mse

should i just post it here

[alpine sable]

maybe mark spoilers

[brisk thunder]

it has to do with 3 not dividing into 100 (!)

[alpine sable]

unsurprising

[brisk thunder]

||https://math.stackexchange.com/q/299651||

[naive valley]

that's slick as hell, imagine having to come up with this in an exam though

[mossy lion]

what level class is this?? like intro linear alg?

[alpine sable]

this feels more like a contest kinda problem than a class kinda problem

the observation you need to make to solve it isn't really linear algebra

[lone heart]

@ivory fern Has your question been resolved?

[ivory fern]

Yes that solves the problem

But ||we have to use a sixth root of unity instead||

[ivory fern]

Yes, it was a math contest

However this problem was dramatically harder than the other three

[ivory fern]

^

[flat dawn]

ah thanks

@ivory fern reading on the solution I see they somehow use 1+w^2 = -w ?

ah okay that's true bc
1+w+w^2 = w(1+w+w^2)
and w!=0,1

[ivory fern]

Yes

Anyway that is not needed in the case of our problem

Because the proof changes at that point

[flat dawn]

it's needed for the first argument

= w(BA-AB) thanks to the above

[ivory fern]

But instead of (A+ωB)(A+ω²B) we use (A+ωB)(A+ω⁵B)

With ω a sixth root of unity

[flat dawn]

why so? the proof as it's given solves the problem

[ivory fern]

No, because in our problem we have a slightly different hypothesis

[flat dawn]

Suppose A and B are real n×n matrices with A^2+B^2=AB. If AB−BA is invertible, prove n is a multiple of 3.

and since n is not multiple of 3, it implies that AB-BA is not invertible, that is det(AB-BA)= 0

[ivory fern]

We have A^2+B^2+AB=0

[flat dawn]

ah right

[flat dawn]

honestly that's an extremely clever solution

I bet it arises through some more heavy math I'm not familiar with

[ivory fern]

Perhaps from something related to the second solution

Or maybe there is a more elementary solution

[flat dawn]

why do they bring in w^n in the solution?

" det(A+ωB)det(A+ωB) is a real number, that is, ω^n * det(BA−AB) is a real number"

[ivory fern]

Because det(kA) = k^n det A

[flat dawn]

ohhh right

[flat dawn]

I don't think it can get more elementary than this, then again it's really hard to come up with

[ivory fern]

But maybe something easier to come up with

[flat dawn]

possibly with high level math which is beyond me lol

[ivory fern]

It would not make sense that we were supposed to find this solution in the time we were given

[ivory fern]

I think we weren't supposed to use too advanced math

Just the most basic things that are studied at the university

[flat dawn]

it might be just someone "showing off" and going "look how simple the solution is" even though it's not

[ivory fern]

I think there might be an easier solution using A^3 = B^3 and something like Jordan normal form. Or maybe not easier but at least easier to come up with

[marsh fog]

I feel completely vindicated reading this.

Beat myself up for like 2 hrs not being able to do this yesterday lol.

[flat dawn]

how would you know that A^3 = B^3

[marsh fog]

They proved it earlier in here.

Scroll up.

[flat dawn]

huh cool

[flat dawn]

this means that

det(A^3) = det(B^3)
det(A)^3 = det(B)^3
and since A,B are over R
det(A) = det(B)

[brisk thunder]

you can replace A with -A

i agree that jordan normal form also looks promising

[flat dawn]

ah true

[lone heart]

@ivory fern Has your question been resolved?

[outer elm]

same

[opaque stratus]

Help!!!!!!!!!!!!😭 😭

[versed bear]

I can help you further if you want

Let me make a picture

[opaque stratus]

wow thx

[lone heart]

@ivory fern Has your question been resolved?

[lone heart]

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[strong summit]

Can someone explain b

[tacit arch]

what part don't you understand

[strong summit]

Idk what do they want exactly

[tacit arch]

compare those two expressions

and find an x

[strong summit]

Urm how do I do that 😭💀

What do those 2 expressions even mean

[tacit arch]

1.03^12 means 1.03 times itself 12 times

[strong summit]

Yeah ik but like how are they related to each other at all

[tacit arch]

1.03 * 1.03 * ... * 1.03 = 1.03 ^ 12

the first one is (1+3x) times itself 12 times

(1+3x) * (1+3x) * ... * (1+3x) = (1+3x) ^ 12

[tacit arch]

taylor expansions give approximiations

[strong summit]

Taylor expansions?

[tacit arch]

oops, binomial

[strong summit]

Approximations of what

[tacit arch]

values like these

[tacit arch]

instead of multiplying by itself 12 times, you can just add a few terms

[strong summit]

Am so lost man😭

[tacit arch]

to get an approximation

if you didn't learn taylor expansions, you don't need to learn it now

[strong summit]

Ik how to find them

Idk what they are tho

[tacit arch]

https://www.khanacademy.org/math/ap-calculus-bc/bc-series-new/bc-10-11/v/maclaurin-and-taylor-series-intuition

[strong summit]

What’s a Taylor 😭💀

Nvmmm all of this back to the question

What’s the relation between the 2

[lone heart]

@strong summit Has your question been resolved?

[lone heart]

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[outer steppe]

@alpine sable , I still don't understand

[alpine sable]

draw a diagram

[hardy eagle]

Can i get help here?

[lone heart]

@outer steppe Has your question been resolved?

[outer steppe]

Yes, in a channel under ✅MATH HELP (AVAILABLE)

[alpine sable]

yeah

[outer steppe]

So how does it prove that the area of the picture is the area of the border?

[alpine sable]

it doesnt prove anything

that piece of information was given in the question

[outer steppe]

I wrote this before

But it says the area of the border = area of the frame

• An 11x16 picture is to be framed. Find the width of the border if the area of the border is equal to the area of the frame. Include a diagram *

[alpine sable]

how do you define border and frame

I would have said that the inside = area of frame

but idk

[outer steppe]

Isn't frame the outer edge of the blue part, and border is the blue area?

[alpine sable]

so are you saying that the border is the same thing as the frame?

[outer steppe]

like this?

[outer steppe]

No the border is shaded

And the frame's area is thr total area?

[alpine sable]

if those two things had the same area, then the inside picture part has area 0

so I dont think that interpretation is correct

it might be better to ask your teacher for clarification

[outer steppe]

Okay, thanks for trying

.close

[lone heart]

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[primal galleon]

Hi everyone,

I hope you're all doing well. I'm currently working on a problem and could really use your help in finding a solution. Here's the equation I'm trying to solve:

1380: A fuel mixture for a certain engine should contain 19 parts gasoline and 1 part oil. Lena needs 25 liters of fuel. She has pure gasoline and a gasoline-oil mixture containing 85% gasoline. How should she proceed?

To tackle this problem, I've come up with a plan. Let's assume Lena uses x liters of the gasoline-oil mixture. In practice, this means she would use 0.85x liters of gasoline and 0.15x liters of oil. Now, Lena needs to mix in y liters of pure gasoline to achieve the right proportions.

By doing so, she will have:

• Used x + y liters of the liquid.
• Used 0.85x + y liters of gasoline and 0.15x liters of oil.

I would greatly appreciate it if someone could guide me on how to proceed from here. Any insights, suggestions, or explanations would be extremely helpful. Thank you all in advance for your assistance!

[proud fjord]

Good solving.
You just have to imput the numbers to get the system of equations.
x+y = (total fuel)
0.85x+y = (total gasoline)
0.15x = (total oil)

Well, there is also the fuel solution ratio that you can write an equation of.

[lone heart]

@primal galleon Has your question been resolved?

[primal galleon]

@proud fjord x + y = 25 (total fuel needed is 25 liters)
0.85x + y = (total gasoline in the mixture)
0.15x = (total oil in the mixture)
Additionally, we can express the fuel solution ratio equation:

x : y = 19 : 1 (fuel solution ratio of 19 parts gasoline to 1 part oil)

Like that?

[proud fjord]

The fuel ration does not look right. x and y is the liters of mixed and pure gasoline respectively.

[primal galleon]

You're right, x represents the liters of the gasoline-oil mixture, and y represents the liters of pure gasoline. Let's correct the fuel solution ratio equation:

(0.85x + y) : y = 19 : 1 (ratio of total gasoline in the mixture to pure gasoline)

[proud fjord]

Yep, now if you write the ratio as fractions youll have 2 equations and 2 variables, making it a solvable system

ratio equation and totals equation

[primal galleon]

Ratio Equation:
(0.85x + y) / y = 19/1

Totals Equation:
x + y = 25

,w (0.85x + y) / y = 19/1; x+y = 25

[ocean seal]

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[primal galleon]

@proud fjord

[proud fjord]

oops, there is a mistake on the ratio equation

A fuel mixture for a certain engine should contain 19 parts gasoline and 1 part oil

[primal galleon]

Ratio Equation:
(0.85x + y) / x = 19/1

Totals Equation:
x + y = 25

So like this instead?

,w (0.85x + y) / x = 19/1;
x + y = 25

[ocean seal]

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Requested by dgh#0553

[proud fjord]

well, x is the ammount of mixed fuel, not the total oil

and the question asks for 19 pure gasoline to 1 pure oil[

[primal galleon]

Ratio Equation:
y / (0.15x) = 19/1

Totals Equation:
x + y = 25

,w
y / (0.15x) = 19/1;
x + y = 25

[ocean seal]

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[primal galleon]

@proud fjord

[proud fjord]

Still, you have to consider that there is gasoline in the mixed fuel too

Refer to the equations of total gasoline and total oil you wrote previously

[primal galleon]

Total Gasoline Equation:
0.85x + y = (total gasoline)

Total Oil Equation:
0.15x = (total oil)

Would this be more accurate?

[proud fjord]

Yes, so you have the equations. The question says that total gasoline to total oil is 19 to 1

[primal galleon]

Total Gasoline Equation:
0.85x + y = 19/20 * (x + y)

Total Oil Equation:
0.15x = 1/20 * (x + y)

[proud fjord]

Yes, that also works

Or the correct writing of the rations.
(total gas) : (total oil) = 0.85x + y : 0.15x = 19 : 1

[proud fjord]

But this is good too

[primal galleon]

,w 0.85x + y = 19/20 * (x + y); 0.15x = 1/20 * (x + y)

[ocean seal]

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[primal galleon]

@proud fjord

[proud fjord]

In the system you just wrote you can sub (x+y) for 25liters to get a numerical answer

The solution you got here is y~~2x

So to get 19 to 1 proportion you put:
2x to 1y
2mixed to 1nonmixed

,w 0.85x + y = 19/20 * (25); 0.15x = 1/20 * (25)

[ocean seal]

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[proud fjord]

and here is the numerical value for 25 liters

[primal galleon]

I see, thanks for the help

[lone heart]

@primal galleon Has your question been resolved?

[lone heart]

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[wraith glade]

Alright so, I've got a trig problem, I

[wraith glade]

hit enter too early

retyping

Alright so, I've got a trig problem, I don't know which formula to use/how to solve it, but I have the word problem:

Angular and Linear Speeds
The circular blade on a saw rotates at 5000 revolutions per minute.

(a) Find the angular speed of the blade in radians per minute.

(b) The blade has a diameter of 7 1/4 inches. Find the linear speed of a blade tip.

For A, I was trying to use the angular speed formula (w = (theta/time) ) but I realised I have no idea what to do with the formula after plugging it in. For B, I'm not totally sure how to proceed yet.

[fierce prairie]

do you know how many radians are in one revolution?

[wraith glade]

like question for me or question about the problem?

If for me: 2pi

[fierce prairie]

for you

yep

[wraith glade]

so i should plug that into the numerator and put 60 in the denominator right?

or am i on the wrong track

[fierce prairie]

so we have 5000 rev / min and if 2pi radians are one revolution, multiply by 2pi radians / 1 revolution

[fierce prairie]

why 60?

[wraith glade]

not sure

for time I believe?

[fierce prairie]

we aren’t worried abt changing the time

[wraith glade]

like 60 seconds

oooh okay

[fierce prairie]

they want to keep it in minutes

[wraith glade]

so the answer is just flat out 2pi per minute?

[fierce prairie]

uh no

[fierce prairie]

maybe read this back

[wraith glade]

2pi / 5000?

actually, I should probably ask what kind of answer I would get from doing this equation

[fierce prairie]

$\frac{5000 revolutions}{minute}\cdot \frac{2\pi radians}{1 revolution}$

[ocean seal]

kihei

[fierce prairie]

this may help to see it

sorry for no spaces lol

but essentially we are “cancelling out” the revolutions unit to end up with radians / minute

[wraith glade]

okay that makes more sense but how does it become 2 fractions instead of one, my notes from the class today only show up with having one fraction somehow

[fierce prairie]

well yeah I didn’t solve it

I’m multiplying those fractions

that’s what the dot is

[wraith glade]

;-;

okay so

[fierce prairie]

this is barely even fractions tho

the denominators are 1

[wraith glade]

i just multiply those and thats that

[fierce prairie]

just multiply 5000 to 2pi

yea

[wraith glade]

damn thats easy

[fierce prairie]

yeah haha I just like to use these sort of fractions as it makes it easier when converting many units

[wraith glade]

so from doing the multiplication I got 31415.9265, this doesn't feel like the final answer

is it the final answer?

oh it is

okay im still stumped with B

wait one sec

i think ive had a break through

[fierce prairie]

so linear speed is just angular speed times radius

[night barn]

I need help

[wraith glade]

wrong channel?

bro its the diameter

i was plugging diameter instead of radius

alrighty

thank you kind math man

.close

[lone heart]

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Use .reopen if this was a mistake.

[wraith glade]

wait

.reopen

[lone heart]

✅

[wraith glade]

how does that work

.close

[lone heart]

Channel closed

Closed by @wraith glade

Use .reopen if this was a mistake.

[fierce prairie]

.you got it or nah?

[wraith glade]

yeah

how does closing the channel work

thats what im confused about

[fierce prairie]

oh lmao it takes a bit

[wraith glade]

oooh

okay

have a good one

[night barn]

so if i need math help where dos one gO?

[wanton valley]

#❓how-to-get-help

[lone heart]

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.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[shrewd panther]

ask

[shrewd panther]

8x^5/(32x^2)

I think the answer is -24x^3 but I highly doubt it

[torn elk]

why -24?

[tacit arch]

can you use parentheses

[torn elk]

not like that

he means like this 8x^5/(32x^2)

[shrewd panther]

Thank you

[torn elk]

ups

[shrewd panther]

no

[torn elk]

i dk why i put 35

but u know what i mean

under?

[shrewd panther]

no

[torn elk]

under what

[shrewd panther]

I'm sorry i haven't slept in a while i'm a bit confused

[shrewd panther]

I tought it is 8-32 = -24x^3

[torn elk]

I'm sure you can do this now

[torn elk]

oh ok

i invented the negative sign lol

so you have this

[shrewd panther]

so you end up with 2^-2 . x^3

[torn elk]

and what's 2^(-2)?

[shrewd panther]

yes ofcourse

[torn elk]

a^(-b) = 1/(a^b)

[shrewd panther]

what

[shrewd panther]

1/4?

[torn elk]

yes

and 1/4 times x^3 is

[torn elk]

done

[shrewd panther]

.close

[lone heart]

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[torn elk]

your painting skills are worse than mine, but ok xD

[shrewd panther]

https://cdn.discordapp.com/emojis/884519673314558022.webp?size=64

[lone heart]

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• After 15 minutes, feel free to ping <@&286206848099549185>.
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[alpine sable]

i do not understand this second step

[alpine sable]

given this

i get this

i dont understand why last det is A_n

[vocal quiver]

Expand by the first column

[alpine sable]

the last det is triangular, meaning its det is the the diagonal

[vocal quiver]

Doesn't look triangular to me

[alpine sable]

?

[vocal quiver]

You have the extra 1 above the diagonal

[alpine sable]

oh

it isnt

but not on the first terms

just on some

anyway, how is that the same as A_n?

[vale crag]

tbf their latexing sucks a lot, they're missing 1's all over the place

[alpine sable]

yeah i just noticed

[vale crag]

but yeah it's expanding on the first col again

your matrix in green should be this

(empty cells being 0s)

if you expand on the first col

[alpine sable]

it would be like this, right?

diagonal 1+a

top of 1 and under a

[vale crag]

you're talking about which matrix ?

the one you circled in green on your pic ?

[alpine sable]

i was talking about the original one

so okey, i see the first det is A_n-1

[vale crag]

yeah so we have the same thing ok

[alpine sable]

Or A_n+1, i dont care

but the second one

[vale crag]

now expand on first column again for the second

[alpine sable]

if i expand by it first row

column?

why not row?

[vale crag]

I mean you can it's faster lol

well yeah you straight up get 1*(det for n-2) then

[alpine sable]

yes

[vale crag]

well then we won

[alpine sable]

well, but thats not the same as here

[vale crag]

wdym

it's exactly what they have

[alpine sable]

and there they have (1+a)det(A_n-1)) - a det (A_n)

oh true

i have the same

i just forgot they were starting at A_n+2

so my n-2 is their n

srry 😄

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

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[winter seal]

aaa

[winter seal]

I keep getting1792pi/15 as my answer when it should be 2048pi/15

My denominator is right but my numerator is wrong

which means that I t hink my limits/boundaries are wrong

or interval sorry

[marsh fog]

What is the actual question?

Are you finding an area between the curves?

[winter seal]

I'm finding the volume

[marsh fog]

Useful to post that in future!

[winter seal]

Sorry!!!

[marsh fog]

(4x)^2=16x^4 no?

Sorry

x^2

Not x^4

[winter seal]

wait a minute

[marsh fog]

Line 2 of your computation of the integral

Think you just forgot to square the 4

[winter seal]

wait

we square the 4 too??

[marsh fog]

(4x)^2 means everything in that bracket squared

[winter seal]

wait omg is that constant not a function ofo that x???

[tardy stag]

why are we integrating from -4 to 0? the intersection points are at x=0,4 right?

[winter seal]

I JUST REALIZED THIS TOO

So

my mistake was instead of setting the two equatiosn EQUAL to each other, I just added them together and made it equal 0

which is not how things go

[tardy stag]

yeah that's not quite it haha

[winter seal]

2nd mistake was that I forgot to square the 4

Okay I'm gonna erase eveyrthing and try again

[marsh fog]

It would be subtracting one from the other that achieves what you want

Hence the sign error there

[winter seal]

yess

[lone heart]

@winter seal Has your question been resolved?

[winter seal]

still doing it

GOT IT

I got 2048

[winter seal]

aoky

okay

on to the enxt one

next*

[lone heart]

@winter seal Has your question been resolved?

[winter seal]

<@&286206848099549185>

[last ether]

You did not change your bounds

You used x-bounds on du

[proven leaf]

if you do a substitution you need to change ur bounds as well

[winter seal]

o.

What if I do a U subsitution of 4+x^2??

[visual girder]

wait is this from the stewarts textbook lol

[winter seal]

This is frommm

It's Calculus Early Transcendentals

11th edition

[visual girder]

ohh

[winter seal]

yeeh

[visual girder]

also for the problem

i would suggest graphing it on desmos

and then seeing if disk or washer would fit it

and then plugging in the values

[visual girder]

oh and shell too

[winter seal]

but wouldn't u determine that by seeing which axis its rotating by?

[visual girder]

hmm

[winter seal]

for example x would be discs

y would be washers

thats what I was taught

[visual girder]

hmm from what ive been taught

idk how to explain this lol

uhh lemme try graphing it and seeing waht we use

[winter seal]

oke

[visual girder]

oke i graphed it

the blue shaded region is wat we are reflecting across the x axis

alright for this one i believe we use a disk

[thorn tapir]

Oh is this about rotation along the marked axis?

[visual girder]

this shaded region is rotated about the x axis

[thorn tapir]

Yee gotchu

[visual girder]

so it forms kinda like a cone pointing to the right i believe

i think we use dy? correct me if im wrong

[thorn tapir]

Either pointy or smooth one of those

[visual girder]

mhm but i dont think it would matter too much

not sure what a is for the integral but** b** is 1.099

[thorn tapir]

Nah I think this is fine

I mean dx

[visual girder]

oh yeah cuz we will be sliding the disk from left to right

so change in x meaning we use dx

[thorn tapir]

See, the solid is made up of infinitesimal disks, if you recall how you integrate sphere's volume thing

You just keep imagining radius that changes with respect to x, then the formula makes sense

[visual girder]

radius would be e^x i believe

[thorn tapir]

Wait is your y, e^x?

Then I think there is no problem in using -infty for a

It really does go until that after all

[visual girder]

this is all the equation of lines @winter seal 's picture had

[thorn tapir]

This improper integral should exist

[visual girder]

this volume problem doesnt seem like it has a nice solution lol

[thorn tapir]

$V = \pi \int_{-\infty}^{1} e^{2x} dx$

[ocean seal]

Riku

[visual girder]

wait how on earth would u solve this

[winter seal]

the solution is

[thorn tapir]

I think it should have one of the nicest solutions

[winter seal]

:)

[thorn tapir]

!nosols

[lone heart]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[winter seal]

oh.

[visual girder]

LMAO

[thorn tapir]

[winter seal]

I worked thruogh the problem

though

here

I'll send my work

[visual girder]

ohh nicee

[winter seal]

?

is that allowewd

[visual girder]

yeah i think it is

[thorn tapir]

Do you know how to integrate e^2x?

[winter seal]

yeah

oh to yes

[visual girder]

ik that part but i meant the part where we plug in infinity lol

[thorn tapir]

You actually usually don't plug in infinity but rather find a limit

This is called improper integral

[visual girder]

ohh ok

i havent learned those yet

i just finished calc 1 like this month lol

[winter seal]

REMEMBER

IF UR TAKING CALC 2

[ocean seal]

Riku

[winter seal]

REMEMBER ALL THE RULES FOR INTEGRALS AND DERIVATIVES

AND THE TRIG IDENTITIES

that's my biggest struggle rn

[thorn tapir]

I feel you I really hate calculus

[winter seal]

I'm tkaiong calc 2 rn

I actually like but I don't remember any identities and stuff

[thorn tapir]

Did you get this? @visual girder

[visual girder]

😭

[visual girder]

nope

[thorn tapir]

oh ;-;

[visual girder]

ig ill udnerstand when i take calc 2

idk why i as a calc 1 person was helping someone taking calc 2 😭

[thorn tapir]

What I mean is, you plug in L like your normally solve

And then find the limit when L goes to negative infinity

Like you do for infinite limits, I think that should be covered in calc 1

[visual girder]

hmm

kinda makes sense

i guess

[thorn tapir]

If you want to know more, you can read about Improper Integrals

But tbh the core definition you'll only see in a Real Analysis course rather than any calc course but they might still teach you this

(yes integration has a lot of theory )

[visual girder]

ill go through this pain later

[thorn tapir]

Take your time

Try what I suggested for now

I think it should work

[visual girder]

rn im redoing the ap calc bc course on khan academy

so i can like remember calc 1 stuff for calc 2

[winter seal]

when r u taking calc 2?

[visual girder]

august!

when school reopens

im in highschool but im taking math at the local community college

[thorn tapir]

damn you guys still in high scho- oh right I did this in my last year of high school also

[visual girder]

but jokes on you im doing this as a sophmore/junior

and senior too i guess

[thorn tapir]

Nothing wrong with that I didn't know m about math in high school

Despite doing everything and getting somewhat good grades

[last ether]

I just graduated fcuk you

[thorn tapir]

[last ether]

Well I did calc 3 my senior year so

But still fuck you

[thorn tapir]

[thorn tapir]

I didn't do any proper course on calc

Honestly I was spared i hate most computations there

Though Several Variables would be interesting to do

[tardy stag]

calc 3 has a lot of really annoying computation

mostly around velocity/curvature/torsion

[winter seal]

calc 3 is multivariable right?

[tardy stag]

yeah

[winter seal]

ew

[proven leaf]

I just learned how to do the Gaussian actually

[winter seal]

imagine having to consider Z into the friggin

kfeokfsepgjf

[thorn tapir]

Guess who has it this starting sem on august alongside Rings and Modules and Number Theory and Differential Equations

I really hope it's mostly theory and not computation

[tardy stag]

the ideas of multivarable made a lot more sense to me in college

[thorn tapir]

I am absolutely trash at Computation

[thorn tapir]

Because you already did some intro analysis amirite

[tardy stag]

yeah true

[thorn tapir]

The guy who would teach us that this sem doesn't care about computations at all so I'm probably saved

[tardy stag]

but i also, like, cared about the theory instead of just the procedures

[thorn tapir]

I only care about the theory 😭

Because I can just look up procedures if I need it

[last ether]

Never had to deal with torsion :)))

[visual girder]

aint no way this turned from a help session to a flexing session

[last ether]

It says that τ = -N • B' which makes sense

[winter seal]

.close

[lone heart]

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[hollow oyster]

Hello, could someone tell me what's the meaning of those n 1, n 2,... n n vertical perantesces?

[raven dagger]

Combinations

Are you fimiliar with the nCr button on calculator?

[hollow oyster]

No I'm sorry, could you give a simple example?

[raven dagger]

So basically in combinations
The top number tells you the total number of things you have
And the bottom number tells you how many things you must chose .
Then the answer to that gives how many possible sets of things you could choose

[hollow oyster]

Cool, thanks for explanation. So (n 0) means we have n of something which we cannot take anyone? and (n 1) means only one from n?

[raven dagger]

Yes

It tells you the number of ways you can do that

[hollow oyster]

Thank you so much for your time, really appreciate it. Math bless you.

.close

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[still wadi]

Can anyone help me with this question

[raven dagger]

Near - 1 is an asymptote

What happens to values around vertical asymptotes

[still wadi]

are you asking me?

[raven dagger]

Yes

[still wadi]

they can become increasingly large?

[raven dagger]

Or small aswell

Can go to really small negative change

[still wadi]

ohh ok

is this true or false

<@&286206848099549185>

[lone heart]

@still wadi Has your question been resolved?

[tardy stag]

what do you think? Can you give an example?

[naive valley]

the question doesn't make much sense - the numerator is a number and yet we're supposed to consider the x values where the numerator is zero?

[lone heart]

@still wadi Has your question been resolved?

[lone heart]

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[still wadi]

can anyone help

[undone falcon]

what do you need help with, you already selected the correct answer

[still wadi]

ok thank you just making sure

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@still wadi Has your question been resolved?

[lone heart]

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[neat oracle]

how to start

[tardy stag]

probably look for a trig sub? I think?

[mellow grail]

Some form of u sub? Also a^2 is a constant since we r integrating with respect to x

[wild trail]

my guess is to take x = a sint

[neat oracle]

thanks it worked

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[dull lance]

How do you do this

Simultaneous equations?

[nimble fern]

,rotate

[nimble fern]

which one is A?

[dull lance]

The lil one

[nimble fern]

no clue

[dull lance]

I have no idea is well

[raven dagger]

Querter of the area of a circle with area x = pi x² /4 = 2Pi x

X = 0 is a solution or

x/4 = 2
X = 8 ?
Are you sure its the little one ?

[dull lance]

Not sure

I thought itw as the lil one

[raven dagger]

Ok now assuming its the larger one

Pi (x+1)² /4 = 2pi X

X² + 2x + 1 = 8x
X² -6x +1 = 0

1/6 isn't a solution tho

Maybe its the narrow strip

[dull lance]

Hmmmm

[raven dagger]

Pi [(x+1)² - x²] / 4 = 2pi X

2x + 1 = 8x

Yes this gives 1/6

A is the small strip between the 2 arcs

[dull lance]

Wowwwwwww

SHIIIIIIIIIIIII

OKAY

ALRALR

THANKS 2 3 5 7

🐐

How did you form the equation though

[lone heart]

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[regal shore]

What will be the HCF and LCM of 0 and 2

[regal shore]

HCF is same as GCD

[vale wigeon]

do you know in general what gcd(0,x) and lcm(0,x) are

[lone heart]

@regal shore Has your question been resolved?

[regal shore]

no

[vale wigeon]

right

yeah so these present a nuance in the definitions of gcd and lcm that basically only comes up when 0 is involved

tho i guess for lcm specifically its a bit easier

since the only multiple of 0 is 0 itself

so theres not much else that lcm(0, x) could be

[regal shore]

0

yea thnx i get that

cuz x*0 is also 0

and what abt HCF?

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[white spear]

Company A has a turnover of 1.5m, and costs of 1.5m.
Company B has a turnover of 0.5m, and costs of 0.1m.

Company A provides goods to Company B, so that the cost of these goods increases Company A's turnover, and Company B's costs.

What would Company A charge Company B for these goods so that both companies have the same percentage profit?
Effectively working out what the green value is when the blue values are equal

[white spear]

@alpine sable 👋

[alpine sable]

Yes

Smh I can't seem to find an exact value wait

0.31355287256600434m

[white spear]

Ah amazing man, thank you

[alpine sable]

Ookay

Do you want the steps?

[white spear]

Yeah please

[alpine sable]

% profit for compA after the transaction

$$
Profit%_{A} = \frac{T_A' - C_A}{T_A'} \times 100% = \frac{(T_A + x) - C_A}{T_A + x} \times 100%
$$

[ocean seal]

weig

[alpine sable]

and for compB its

$$
Profit%_{B} = \frac{T_B' - C_B'}{T_B'} \times 100% = \frac{T_B - (C_B + x)}{T_B} \times 100%
$$

[ocean seal]

weig

[alpine sable]

since new cost ( C_B' ) is ( C_B + x ) and the turnover ( T_B' ) remains the same as ( T_B )

[ocean seal]

weig

[alpine sable]

and we want the two profits to be equal

so we gotta solve for x

$$
\frac{(T_A + x) - C_A}{T_A + x} = \frac{T_B - (C_B + x)}{T_B}
$$

[ocean seal]

weig

[alpine sable]

for the calculation

let me just scan my tablet i didnt write this on ltx

oh nvm i can transfer

$$
\frac{(T_A + x) - C_A}{T_A + x} = \frac{T_B - (C_B + x)}{T_B}
$$

[ocean seal]

weig

[alpine sable]

wait no

ah yes

$$
\frac{(1.5m + x) - 1.5m}{1.5m + x} = \frac{0.5m - (0.1m + x)}{0.5m}
$$

[ocean seal]

weig

[alpine sable]

simplifying you get

$$
\frac{x}{1.5m + x} = \frac{0.4m - x}{0.5m}
$$

[ocean seal]

weig

[alpine sable]

to clear the fraction by cross multiplying you get

$$
x \times 0.5m = (0.4m - x) \times (1.5m + x)
$$

[ocean seal]

weig

[alpine sable]

and from here you want to expand the right side of the eq

$$
0.5m \times x = 0.4m \times 1.5m + 0.4m \times x - x^2 - 1.5m \times x
$$

[ocean seal]

weig

[alpine sable]

simplify again

$$
0.5m \times x = 0.6m^2 - 0.6m \times x
$$

[ocean seal]

weig

[alpine sable]

rearranging you get

$$
1.1m \times x = 0.6m^2
$$

[ocean seal]

weig

[alpine sable]

$$
x = \frac{0.6m^2}{1.1m = ?

$$

[ocean seal]

weig
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[alpine sable]

$$
x = \frac{0.6m^2}{1.1m = 0.31355287256600434m

$$

why bro

[ocean seal]

weig
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[alpine sable]

what