help-13

[snow timber]

Hello, I need some help digesting what has been done here. Not in the details of the proof, but rather in its semantics.

I read the proof as 'y(x) is a solution to the inhomogeneous linear ODE iff y(x) can be written as y(x)=yc(x)+yp(x), where yc(x) solves the corresponding homogeneous linear ODE'. This is equivalent to 'If y(x) is a solution to the inhomogeneous linear ODE, it can be written as y(x)=yc(x)+yp(x),where yc(x) solves the corresponding homogeneous linear ODE ' but also 'If y(x) can be written as y(x)=yc(x)+yp(x) where yc(x) solves the corresponding homogeneous linear ODE, then y(x) is a solution of the corresponding inhomogeneous ODE'.

My confusion is in which direction the attached image actually takes for the proof? Because they start by saying 'If y(x)=yc(x) + yp(x) is a solution' but surely only if they were proving the forward direction they would assume y(x) to be a solution since the point of the other direction is to say that if y(x)=... then it is a solution. However, they've also said that y(x)=yc(x)+yp(x), which surely constitutes the other direction...

[snow timber]

Sorry I accidentally clicked enter

[dire geode]

is that the complete proof

[snow timber]

This is the last line sorry

[dire geode]

yes the "if and only if" is complete then with that

[snow timber]

So does their proof go both ways? Can I have clarification on which direction it proves in which order

[dire geode]

helps to give variables to the two statements

e.g. P if and only if Q

P = "y(x) is a solution of (3.3)"
Q = "y(x) can be written as y(x) = y_p(x) + y_c(x) .... complementary function"

[cedar kiln]

@snow timber Has your question been resolved?

[snow timber]

Yeah and then break that up into if P then Q and if Q then P right? and prove the forward direction followed by the backwards direction or vice versa?

Assuming that's correct, my main question is how have they said 'If y(x)=yc(x)+yp(x) is a solution to (3.3)' at the beginning? Like isn't that assuming both parts i.e they assume that y(x) is a solution and that it can be written in that way?

[snow timber]

My belief now is that in the picture they prove the backwards direction, that is that if y(x)=yc(x)+yp(x) with yc(x) .. then y(x) is a solution, and that I am potentially misunderstanding their phrasing? I think by asserting 'If y(x)=.. is a solution to (3.3)' I thought they were assuming it was a solution, but actually they are just being speculative and saying 'if this was true then it should = f(x) when plugged into the ODE' and plugging it in to show that's true.

.close

[cedar kiln]

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[teal frost]

Question

[teal frost]

Which one?

When its negative is it =/> 1 or =/< 1

Or should the 1 also change I'm honestly not sure how the | | works here

[civic pumice]

depends where x lives

|x| = x if x >= 0 and |x| = -x if x <0

[modern igloo]

if you are talking about that absolute value thing it would unwrap as
$$
-1 \leq \frac{x+1}{x-1} \leq 1
$$

[civic pumice]

lool

[teal frost]

Oh so its

[dusk goblet]

get rid of ||

[wraith dagger]

Armagidon

[modern igloo]

yeah, thanks

[teal frost]

Like this?

?

[modern igloo]

yeah, pretty much. And that forms a system of inequalities

[wicked mantle]

absolute value

[teal frost]

Thanks

.close

[cedar kiln]

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[blazing dune]

given a number n, is there a faster way (other than brute force) to find the number of pairs (a, b) where b is divisable by a such that a + b = n

[blazing dune]

yeah the number of pairs not the pairs themselves

[azure swift]

Uh well if b is divisible by a then there exists k such that b = ka, so ur equation is a + ka = n

[crimson delta]

so b=am, (m+1)a=n ?

aka all divisors of n ?

[civic pumice]

Should be able to find a property that n satisfies from this..

[high bridge]

^ yeah that is it

just find all factor pairs of n and you can find (a,b)

[azure swift]

It’s just this

[blazing dune]

ok, so I just need to find the divisors of n.
that makes very sense
I don't know how I didn't think of it.
thanks all

.close

[cedar kiln]

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[cedar kiln]

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[manic zephyr]

I need help with operations of functions

[tropic oxide]

progress?

[manic zephyr]

none, im just starting

[tropic oxide]

do you know that you are asked for g(1)*h(1)

[manic zephyr]

yes

[tropic oxide]

ok

do you see what to do

[manic zephyr]

I substitute them

[tropic oxide]

ok that sounds like you probably have the right idea

so do it

[manic zephyr]

im not sure if im right

[tropic oxide]

show your work

[manic zephyr]

(n^2 + 4 + 2n)(-3n + 2)

then i substitute n=1?

[dusk finch]

that works, I'd do it right now though, without expanding it

[tropic oxide]

emphasis on without expanding

[manic zephyr]

is it going to be like 1^2 + 4 + 2(1) + -3(1) + 2?

[tropic oxide]

no

[pastel vault]

they closed the same question then asked again

[tropic oxide]

you do not replace the multiplication with an addition all of a sudden

[tropic oxide]

bruh.

[manic zephyr]

no one was answering me earlierr

[tropic oxide]

!noclopen

[cedar kiln]

Please don't repeatedly close and claim a new channel with the exact same question. This erases all previous progress made towards your problem and is confusing for helpers, making it more difficult to help you. Please be patient, even if your channel has not received much attention.

[pastel vault]

to be fair she didn't get the idea behind g(1) * h(1) at all that time

[tropic oxide]

she*

[pastel vault]

a lot of people were...

[manic zephyr]

yea but I asked earlier on how to simplify it yet no one answered me after a few minutes

[tropic oxide]

anyway, this is incorrect

[manic zephyr]

how can I correct it?

[tropic oxide]

by putting the multiplication back where it belongs

i.e.

(1^2 + 4 + 2*1) * (-3*1 + 2)

and work this out

[manic zephyr]

ohh

[tropic oxide]

do you understand why just randomly replacing a times sign with a plus sign is no good

[modern estuary]

Thats what substituting n=something means

[manic zephyr]

do I do the FOIL method here?

or just solve it as is?

[tropic oxide]

.

[tropic oxide]

.

do not expand or "FOIL" here

[manic zephyr]

is the answer -7?

[tropic oxide]

there's no point to it; you only create more work and more chances to screw up

yes the answer is -7

[manic zephyr]

is this the operation of multiplication?

[tropic oxide]

wdym "this"

[manic zephyr]

this onee

[tropic oxide]

well.. yes

the dot between g and h does mean multiplication

[manic zephyr]

how abt for this one, this is for subtraction

do I just do the same thing here?

[tropic oxide]

kinda.

the logic is the same, but the operation is different.

[manic zephyr]

I'll try to solve but can u check if its correct

[tropic oxide]

sure.

[manic zephyr]

(4x - 3) - ( x^3 + 2x)
(4(4) - 3) - (4^3 + 2(4))
13 - (12 + 8)
13 - 20
=-7

am I correct?

[spring forum]

wait, no

4^3 ≠ 4 • 3

[manic zephyr]

oh right

i forgot mb

[spring forum]

np

[manic zephyr]

wait let me try it again

what do I solve first 4x4 or 4^3 ?

[spring forum]

it doesn't really matter, because they're both in different parentheses. as long as you solve the things in the parentheses before subtracting, you're all good

so in the 13-(12+8) step

you got the value in the first parentheses correct

[manic zephyr]

okayy, let me re-solve itt

[spring forum]

you only need to reevaluate the second

[manic zephyr]

is it 181?

[spring forum]

how did you get this value?

I don't think it's correct

[manic zephyr]

oh noo

I solved it as is

can u help me how to solve itt

[spring forum]

can u show me ur steps?

u can also take a picture

[manic zephyr]

(4x - 3) - ( x^3 + 2x)
(4(4) - 3) - (4^3 + 2(4))
(16 - 3) - (64 + 8)
13 - 72
= -59

[manic zephyr]

I realized my mistake hehe

I tried solving it again but in detailed

is it correct?

[spring forum]

It is

[manic zephyr]

omggg

[spring forum]

[manic zephyr]

can u help me w my homework :(

[spring forum]

sure!

[manic zephyr]

yipee thank u so muchh

[spring forum]

one problem at a time tho

[manic zephyr]

ofc

[spring forum]

-# I might have to go soon tho, maybe in 10-15 mins

[manic zephyr]

oh, its okayy

(3x + 3)+(-4x + 1)

?

[spring forum]

Wait, we want (h+g)(10) tho

not (h*g)(10)

it's kind of the same idea as the previous two problems

write h+g out, and then plug 10 in

[manic zephyr]

oh so like ?

[spring forum]

like what?

[manic zephyr]

this

[spring forum]

ah

yes

[manic zephyr]

oh okay

[spring forum]

wait

you could technically plug 10 in

(right away)

but you can make it easier for yourself and simplify

[manic zephyr]

oh okayy

[spring forum]

make it like ax+b

[manic zephyr]

(3(10) + 3)+(-4(10) + 1)?

33 + -41

=-8?

[spring forum]

No, remember that -40+1 ≠ -40-1

[manic zephyr]

oh right

I forgot my integers

is it =-6?

[spring forum]

yes!

[spring forum]

a minor note:

what I was mentioning was, you can make it

(3x+3)+(-4x+1) = (3x-4x)+(3+1)=-x+4

[manic zephyr]

so I can just go straight to the point?

[spring forum]

that would make it much easier; -10+4=-6

[spring forum]

sort of, yes

[manic zephyr]

okayy, I think im slowly understanding it now

[spring forum]

great!

If you're done, you can .close this channel

[manic zephyr]

okayy, I perhaps I need to cuz u need to go

[spring forum]

mm, not yet

[manic zephyr]

oh rlly

[spring forum]

and also, even if I'm not here, other ppl will

[manic zephyr]

okie

this is different

how can I start this

[spring forum]

not much different

you can write it out just as before

try that; if you need help you can ask

[manic zephyr]

okay

is this correct?

[spring forum]

yeah!

[manic zephyr]

I apologize for the way I write, its hard to draw in the pc T_T

[spring forum]

you're good

no need to apologize

[manic zephyr]

8/2

=4

?

[spring forum]

Wait, try again

What is 3*3+2?

2*3-4?

[manic zephyr]

oh

woops

[spring forum]

issok

[manic zephyr]

is it 11/2?

[spring forum]

mhm!

[manic zephyr]

5.5?

[spring forum]

yep

[manic zephyr]

I find this confusing

[spring forum]

here, f(2) is the input of g

$g( \tiny f(2) )$

[wraith dagger]

astraea 💫

[spring forum]

is this going to work

no

so like $g( \color{red} f(2) \color{black})$

[wraith dagger]

astraea 💫

[spring forum]

the red part is what you plug into g

[manic zephyr]

hmm wait let me try

[spring forum]

you can treat it as any other number

just find f(2), and then plug into g

[manic zephyr]

3n + 2 (2)?

[spring forum]

wait, we'll take it slowly

if we let y=f(2), g(f(2))=g(y)

so, you need to calculate y first

and then calculate g

So to start, @manic zephyr , can you calculate f(2)?

[manic zephyr]

hmm let me try..

(3n + 2)((2n^2+5)2)?

[spring forum]

wait, wait, slow down

calculate f(2) first

[manic zephyr]

oh okay

[spring forum]

think of f and g as two machines

we plug 2 into f

and then take what comes out of it and put it into g

[manic zephyr]

there's no substitution happening here right?

for now

[spring forum]

yes, for now

don't worry about g(f(2)), just calculate f(2)

[manic zephyr]

4n^4+20n^2+25??

[spring forum]

Where are you getting that?

[manic zephyr]

wait

[spring forum]

I only asked you to do f(2)

[manic zephyr]

oh wait i think i made a mistake

[spring forum]

don't think about g for now

if f(n)=2n^2+5, what is f(2)?

@manic zephyr

[manic zephyr]

is it 4n^2 + 10?

[spring forum]

remember, f is a machine. We're putting 2 in, which means we're letting n=2

[manic zephyr]

so 2(2)^2 + 5?

[spring forum]

yes!

[manic zephyr]

ah

omg

is this the new f?

[spring forum]

No, it's a machine

for example, think about it as a broken calculator

when f is a broken calculator, and we plug a random number called n into it, it'll return 2*n^2+5

so that means, when we punch 2 into that calculator, we'll get 2*(2)^2+5

does that make sense?

[manic zephyr]

yes, I understand it

[spring forum]

OK

now g is a different broken calculator

and we decide to put whatever we got from f

into g

so we got 2(2)^2+5 from f

you can calculate this out

and then punch the new value into g

[manic zephyr]

no substitution right?

[spring forum]

mm, no

so can you first calculate 2(2)^2+5?

what is this value equal to?

[manic zephyr]

21?

[spring forum]

not exactly…

what's 2(2)^2+5?

[manic zephyr]

oh.

do I solve it?

[spring forum]

not yet, just calculate 2(2)^2+5

because then we'll know what f(2) is

[manic zephyr]

13?

[spring forum]

yes!

So now we know that f(2)=13 — this is the value we got from the f broken calculator

Now we need to put 13 into the g broken calculator

because g(f(2)) is now g(13)

can you calculate g(13) if you know that g(n)=3n+2?

@manic zephyr

[manic zephyr]

42?

[spring forum]

close, we're adding 2, not 3 tho

-# or is that a typo?

[manic zephyr]

41

WOOPS

[spring forum]

Yeah!

So that's your answer

[manic zephyr]

oh

wait, just to be clear

[spring forum]

mhm?

[manic zephyr]

f(n) = 2n^2 + 5
f(2) = 2(2)^2 + 5
=13

?

[spring forum]

mhm!

[manic zephyr]

oh okayy

[spring forum]

-# broken calculators work that way

[manic zephyr]

AHAHAHA

how about for this one

it doesn't give me whats a

[spring forum]

this looks pretty similar to this one :o

[manic zephyr]

yupp

[flat mica]

so the answer will be an expression with a

[spring forum]

@manic zephyr so do the same thing as you did with the previous one, but simplify instead of plugging things in

note that, (g-f)(a)=g(a)-f(a)

and you know the values of both already

[manic zephyr]

hmm okay wait

wait how will I solve this :(

[spring forum]

Although it's complicated, it's actually not

if I let x=g(a) and y=f(a)

then x=-3a-3

and y=a^2+5

the problem is asking for g(a)-f(a)=x-y

can you calculate that?

@manic zephyr

[manic zephyr]

so the answer is not a whole number?

[spring forum]

The answer is a new expression

we won't be getting a number

we'll get a new function, let's say h(a)

So now, if you know that $x=-3a-3$ and $y=a^2+5,$ can you write out $x-y?$

[wraith dagger]

astraea 💫

[spring forum]

@manic zephyr

[manic zephyr]

(-3a - 3)-(a^2 + 5)?

[spring forum]

Yes!!

[manic zephyr]

do I solve it?

[spring forum]

simplify it, yes

open the parentheses and simplify

[manic zephyr]

a^2 + 5?

[spring forum]

hmmm, where did the -3a go?

and also, where did the -3 go?

[manic zephyr]

oh wait I forgot I mixed up my signs

[spring forum]

Start by getting rid of the parentheses (carefully tho, you have -(a^2+5))

[manic zephyr]

is it -6a^2 + 5?

[spring forum]

mmm…no

[spring forum]

^^^

@manic zephyr ?

[manic zephyr]

im confusedd

is it -3 (a+1)?

[spring forum]

when you have (-3a - 3)-(a^2 + 5)

first try to get rid of the parentheses

we can break open the first one, because there isn't a negative sign in front of it:

-3a - 3-(a^2 + 5)

but how do you break open the second one?

@manic zephyr

[manic zephyr]

hold on

i don't know how

[spring forum]

really?

[manic zephyr]

yea..

[spring forum]

do you know how to do -(a^2+5)?

that's ok

[spring forum]

^^ ?

[manic zephyr]

i also don't know

[spring forum]

ok!

so -(a^2+5)=(-1)*(a^2+5)

do you agree with this?

[manic zephyr]

yes?

[spring forum]

what's that question mark for? if you have a question pls ask

[manic zephyr]

im not sure with my answer T_T

[spring forum]

hmm

well -3=(-1)*(3)

do you agree?

[manic zephyr]

yes

[spring forum]

so likewise, -(a^2+5)=(-1)*(a^2+5)

now do you know how to distribute?

[manic zephyr]

nooo

[spring forum]

ok

[manic zephyr]

thats what confuse me the most

[spring forum]

so given a(b+c), we can do a*b+a*c

multiply a into all of the numbers within the parentheses

[spring forum]

so here

we can multiply -1 into all the numbers

(-1)*a^2+(-1)*5

can you simplify this?

[manic zephyr]

let me try

((a^2)-1) + ((-1)*5?

[spring forum]

what is (-1)*x?

for any x

[manic zephyr]

-1x?

[spring forum]

yes!

so (-1)*a^2 is?

[manic zephyr]

-1a^2?

[spring forum]

wait a sec, -1x can also be written as -x

[spring forum]

so this is also?

[manic zephyr]

no

?

[spring forum]

oh yes, it is right

but you don't need to write the 1

-1x=-x

[manic zephyr]

so -a^2?

[spring forum]

like, (-1)*3=-3

[spring forum]

Yes!

And (-1)*5 is?

[manic zephyr]

-5

so is the answer -a^2?

[spring forum]

[spring forum]

wait no

[spring forum]

this is right

so when you expand -(a^2+5)

you get -a^2-5

[manic zephyr]

hm

[spring forum]

when you put it back into -3a - 3-(a^2 + 5)

you get -3a-3-a^2-5

now can you simplify this?

[manic zephyr]

i'll try

(-3a - 3)-(a^2 - 5)

?

[spring forum]

why bracket it up again

[manic zephyr]

how do u do itt

[spring forum]

-3a-3-a^2-5=-3a-a^2+(-3-5)=?

[manic zephyr]

can u do it like this

I understand it that wayy

[spring forum]

-3a-3-a^2-5 is currently x-y, wdym?

[manic zephyr]

no, like put it in a text way

[spring forum]

$-3a-3-a^2-5=-3a-a^2+(-3-5)$

[wraith dagger]

astraea 💫

[spring forum]

which is equal to?

calculate the thing in the parentheses

[manic zephyr]

-8

[spring forum]

yep

so now we have $-3a-a^2-8$

[wraith dagger]

astraea 💫

[spring forum]

can you rearrange it so that a^2 is in the front?

after you do that, we have the answer

all this while, we were just simplifying things

[manic zephyr]

is it -11a^2?

[spring forum]

just rearrange a^2 to the front

we can't merge a^2 and a

[manic zephyr]

wdymm

[spring forum]

we'll just move a^2 to the front

like this:

$-a^2-3a-8$

[wraith dagger]

astraea 💫

[manic zephyr]

oh

[spring forum]

mhm

[manic zephyr]

so -11a^2?

[spring forum]

we can do this because addition is communtative remember

[spring forum]

no, this is a polynomial

we can't keep simplifying

because a^2 ≠ a ≠ 1

I gtg now, if you're done you can .close this channel

[manic zephyr]

wait so what's the answerr

[cedar kiln]

@manic zephyr Has your question been resolved?

[cedar kiln]

@manic zephyr Has your question been resolved?

Channel closed

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[slender ginkgo]

can u use trig?

[fast verge]

For the left triangle:
tan(54°) = 87 / a → so a = 87 / tan(54°) ≈ 63.21

For the right triangle:
tan(23°) = 87 / b → so b = 87 / tan(23°) ≈ 204.96

Then I just added them:
x = a + b ≈ 268.17

[tropic oxide]

uhhh

[fast verge]

I watched the video my instructor gave me but it’s a bit different as he solved for height

[tropic oxide]

x is the DIFFERENCE though

maybe we should be more precise

let me draw a picture with names

[fast verge]

Thanks boss

[tropic oxide]

AB is a bit wobbly but hopefully this is not an obstacle

[fast verge]

So

Solve then subtract

Just a sec

I got 141.75 I’m gonna try to test it

Wonderful!

Thank you!

.close

[cedar kiln]

Channel closed

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Use .reopen if this was a mistake.

[cedar kiln]

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[glossy fossil]

Hey

[glossy fossil]

Design an Non-Inverting Schmitt Trigger Circuit with ±12V supply to have UTP=3V and LTP=0V using Bipolar OPAMP. Draw the hysteresis diagram & explain the operation.Design an Non-Inverting Schmitt Trigger Circuit with ±12V supply to have UTP=3V and LTP=0V using Bipolar OPA MP

[dire geode]

sir this is a math server

[glossy fossil]

It's is math

[slender ginkgo]

this is oddly specific

[rapid mauve]

that's engineering

[glossy fossil]

Yeah

[rapid mauve]

i don't think we are asked to design circuits in math?

unless you mean in graph theory

[dire geode]

nah

[rapid mauve]

but we don't deal with voltages in graph theory

[dire geode]

#old-network for EE server

[dusky panther]

"it's is math" 🥀

[dire geode]

#old-network message

[glossy fossil]

Sorta
I have the reference it's just the 0 thing I am not understanding

[rapid mauve]

then you need to show the math part of the problem

[dusky panther]

#old-network message

[rapid mauve]

the question, as it stands, is an EE question

[glossy fossil]

Okay

[cedar kiln]

@glossy fossil Has your question been resolved?

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[hollow saffron]

Can someone help me with this question?-How many ways can you pick three different numbers from 1 to 1999 so that any two numbers in the chosen set are at least 9 apart?

[hollow saffron]

All the ai's getting am enormous answers, but im getting 71732

[cedar kiln]

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

[chrome fable]

Even i got 71732

[hollow saffron]

Thank you this made me even more confident that i got the right answer

[ocean sentinel]

How did you get 71732?

[hollow saffron]

this set is the intersection of 1-10,2-11,3-12…1990-1999. Then if we only count possibilities where the number “1” is among, we gotta take look at the set 1-10 because the difference gotta be 9. To calculate this, this is just C9 choose 2. This would happen 1991 times and then we finally have the set: 1992-1999. this is just C8 choose 3. Now we add all the stuff together, and that simplifies to 71732.

[ocean sentinel]

But you're only picking three numbers right? not 1991 numbers?

I can't lie that didn't make much sense to me

Fwiw my answer is much much bigger

[hollow saffron]

Yeah only 3 numbers

[ocean sentinel]

How would you use that method to calculate the same thing but only for numbers 1 to 20?

[crimson delta]

just intuitively speaking, given that you can pick from numbers 1 to 1999, the restriction that they are 9 apart is barely a restriction at all

[ocean sentinel]

Exactly

[crimson delta]

so most options should be allowed

so your result should be much closer to 2000C3

which is massive

[civic pumice]

Dont use AI to verify your answers...

[crimson delta]

!nogpt

[cedar kiln]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[hollow saffron]

I think its the same thing, consider 3 element subsets that only contain the number 1, and the restriction is that you can go further than 10, from 9 different things, the number of ways you can choose 2 things is C(2,9) and this continues till {11,12...20} then plus C(3,9) and add all the stuff up

I dont see any illogical stuff here thoo

[ocean sentinel]

So that's (9 choose 2) * 11 + (9 choose 3) then?

480?

The actual answer is 4

[hollow saffron]

There is 2,5,6

6,0,8

6,8,4

3,8,5

10,7 13

[ocean sentinel]

2 and 5 are 3 apart

Not 9

[hollow saffron]

Oh.... No no at least 9 apart

[ocean sentinel]

none of these numbers are at least 9 apart

[rapid mauve]

the only sets that fulfill depression's question are
1, 10, 19
1, 10, 20
1, 11, 20
2, 11, 20

[ocean sentinel]

Have another think about the question and see what you get. But I hope you understand why your method doesn't really work

Like denascite said, it should be in the order of 2000C3

[hollow saffron]

Can you explain why?

[ocean sentinel]

Wait actually... first, when you say "at least 9 apart" do you mean "at most 9 apart"?

Then your answers would make a lot more sense

[hollow saffron]

Oh misread the question

you're actually right

[ocean sentinel]

Lol

[opal hinge]

Let's try to solve depression problem with 1-20 and extend it to general.
We have 3 numbers a<b<c

20>=c
c>b+8
b>a+8
a>=1

Now rewrite it we have 20>=c>b+8>a+16>=17

It now become choose 3 number in range 17 to 20

Which is 4C3=4

[ocean sentinel]

I would just like to say you're basically telling him the answer

[opal hinge]

Oops

[civic pumice]

actually no lol

mb im dumb team

[ocean sentinel]

sad

[civic pumice]

yeah, wouldve been handy

[cedar kiln]

@hollow saffron Has your question been resolved?

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[flint marten]

Triangle ABC is shown below:

Triangle ABC. Line passes through points D, B, and E.

Given: ΔABC

Prove: All three angles of ΔABC add up to 180°.

The flowchart with missing reason proves the measures of the interior angles of ΔABC total 180°:

Top path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle EBC is congruent to angle BCA. By Substitution, the sum of the measures of angles BCA, CBA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Alternate Interior Angles, angle DBA is congruent to angle BAC. By Substitution, the sum of the measures of angles BCA, BCA, and BAC equals 180 degrees. Next path, by Construction, line segment DE is parallel to line segment AC. By Definition of a Straight Angle, the measure of angle EBD equals 180 degrees. By Substitution, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees. Bottom path, by Construction, line segment DE is parallel to line segment AC. By space labeled 1, the sum of the measures of angles EBC, CBA, and DBA equals the measure of angle EBD. By Substitution, the sum of the measures of angles EBC, CBA, and DBA equals 180 degrees.

Which reason can be used to fill in the numbered blank space?

[flint marten]

(I got A, but im not sure. Also, sorry if you cant see 😭 )

[slender ginkgo]

i think its trying to use the fact that angles are added up tgt to form a bigger angle

[flint marten]

its asking what the blank is

heres a better image

[slender ginkgo]

i cant give u the direct ans

[slender ginkgo]

thats the hint

for ref

associativity is grouping property

and commutativity is exchanging property

[flint marten]

exchanging?

[slender ginkgo]

like

a + b = b + c

[flint marten]

o

[slender ginkgo]

switching position

[flint marten]

okay its angle addition postulate

i didnt see that as a answer choice

i js saw it when u gave me the hint 😭

right??

@slender ginkgo

[slender ginkgo]

👍

[flint marten]

can you help? i have an idea how to do this one but im still stuck 😭

[slender ginkgo]

literally the angle bisector

also its an isosceles

[flint marten]

well ik its iscoceles

i need a second measure

like 44 and smth else

[slender ginkgo]

second measure for which angle?

[flint marten]

to do the thing

idk how to explain it

like

180 - 48 - x =

[slender ginkgo]

where did the 48 come from

[flint marten]

oh

its 44

mb

😭

my brains a lil foggy bru

[slender ginkgo]

have u used the angle bisector stuff

u can also just use exterior angle theorem

XAF and XCF are both 44/2, right?

[flint marten]

i forgot them

they were in the last unit tho

[flint marten]

wym

[slender ginkgo]

.

[flint marten]

i uh

idk man

[slender ginkgo]

whats the meaning of an angle bisector

[flint marten]

basically, line that splits in 2 equal parts

[slender ginkgo]

splits angles into 2 parts

[flint marten]

ohh

yesyes

thats what i meant

[slender ginkgo]

so this

[flint marten]

wait

c is apart of bca

nah nevermind

wait

yeahh wait

a is also apart of bca

which is 44

so thats why xaf and xcf is 44 as well??

where u get 44

[slender ginkgo]

44/2

not 44

definitely not 44

[flint marten]

44/2??

so 22???

[slender ginkgo]

yuh

cuz its angle bisector

😭

[flint marten]

it bisects bca?

which is why its 22 on each??

[slender ginkgo]

the line bisects bca in 2

so the angles formed by the bisection are half of 44

[flint marten]

answers 44 then?

why did it take me this long to grasp what ts is bru

i might need a break bru 😭

[slender ginkgo]

https://tenor.com/view/leonardo-dicaprio-clapping-clap-applause-amazing-gif-16078907558888063471

44, indeed

[flint marten]

i am a dumbass

i js forgot some of the stuff from previous units

[slender ginkgo]

we all do

its normal

[flint marten]

alr

[flint marten]

is this right

[slender ginkgo]

,w 180 - (2 * (180 - 104))

[wraith dagger]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by nuclearinteraction

[flint marten]

uh

28 aint a answer

💔

[slender ginkgo]

how did u get 78

[flint marten]

all triangles have to equal 180

i just plugged in the numbers

and to prove it, i just subtracted 180 to 104

to find 78

if that makes sense

[slender ginkgo]

78 is one angle

how many angles are there

[flint marten]

its not 78 mb

its 76

[flint marten]

3

[slender ginkgo]

two of which are?

[flint marten]

wdym

btw i think its 38 now

[slender ginkgo]

76, right?

[flint marten]

no only 1 angle is given

which is 104

[slender ginkgo]

i mean inside the triangle

[flint marten]

180-38-38=104

[flint marten]

wdym inside

im slow

[slender ginkgo]

NOM

[flint marten]

idk man

mop is 104

[flint marten]

yeah it would be 76

[slender ginkgo]

so whats the third angle r

[flint marten]

but if u add those

its 152

i think

and any answerr choice there

would be over 180

so i lowk dont know

[slender ginkgo]

top angle + 152 = 180

[flint marten]

38??

[slender ginkgo]

what is 180 - 152

[flint marten]

28

[gloomy mantle]

its asking for angle nmo only, which is 76 because of isoceles property.

[flint marten]

but it aint a answer choice

[gloomy mantle]

it isnt asking for the top angle (28)

[slender ginkgo]

oh shii

mb mb

76 is indeed correct

plz forgive me

🙏

i thought itd asking for the top angle

[flint marten]

oh

i thought it wasm 38

[gloomy mantle]

noo haha a triangle with 2 congruent sides also has 2 congruent angles. since the right angle is 76 then the left angle must also be 76

but yes you were correct

[flint marten]

alright

tysm guys @gloomy mantle and @slender ginkgo

yall goeated

[slender ginkgo]

once again

mb

for misreading the q

and wasting ur timee

🙏

[flint marten]

na bru

u didnt

i had a hard time understanding lol

.close

[cedar kiln]

Channel closed

Closed by @flint marten

Use .reopen if this was a mistake.

[flint marten]

.reopen

[cedar kiln]

✅

[flint marten]

what were u gonna say?

ok well nvm then

.close

[cedar kiln]

Channel closed

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[cedar kiln]

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[median spire]

what are the prerequisites for linear algebra? is algebra 2 enough?

[slender ginkgo]

some set theory should be useful

[median spire]

as in knowing the basic set operations? or is there more to set theory?

[slender ginkgo]

knowing set, subset, and set operations should be enough for set theory for linear alg imo

but ive just started learning linear alg

[median spire]

i'll see if i find a book in my school's library then and i'll study some when i have spare time from ollympiads lol, thanks btw

.close

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[wintry latch]

Say x is directly proportional to y and to z and is inversely proportional to w. I get tbat x/y and x/z and xw are constant values. But then the book says that (xw)/(yz) being a constant follows from the above. How?

[buoyant latch]

If x/y is a constant, is y/x a constant?

[wintry latch]

Yes

[buoyant latch]

Hmm that at least gets us y/z is a constant

[wintry latch]

O

And then the product of that and xw gives me that constant quality

[buoyant latch]

Well it’s not quite right

That’s xwy/z

[wintry latch]

Oh I see

[buoyant latch]

I’m thinking that 𝛂z = x = 𝛃/w

That makes z inversely proportional to w

Hmm I’m not sure I’d have to play around a bit more

[wintry latch]

So it's just fiddling around until you get constants in terms of x y w and z such that you get the right constants that would then give the required product?

[buoyant latch]

Yeah

[buoyant latch]

This might help

If you just put some scaling numbers on them so it’s equations

Also it feels like something has been cancelled away in xw/yz

[gritty viper]

x = αy = βz = γ/w

Surely this is just not true

x=y=z=w=1
xw/yz = 1

x=y=z=2, w=1/2
xw/yz = 1/4

[civic pumice]

With what you said, (xw)/(yz) = c/x^2 (where c constant ofc)

so its false that its constant, if x isnt

[buoyant latch]

Can you take a picture of the book

[wintry latch]

ok

[buoyant latch]

,rcw

[buoyant latch]

Ok that’s not what they said

They meant it as if you put in actual numbers then the entire thing is a constant

Not that it’s constant for any and all arbitrary values

[wintry latch]

Hmmm

So if I plug on for w and for z random numbers then it becomes x/y times whatever i got for w/z and two constants make a constant

[buoyant latch]

You see how the entire thing depends on some scaling factor and x^2?

[buoyant latch]

That’s what they’ve done here

When (w, y, z) = (6, 8, 5) means x = 4

This info can be used to solve for a constant

Then when you’re given (4, 10, 9) you can use that constant to find x

[wintry latch]

I think I might get it

Thanks for the help

.close

[cedar kiln]

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[cedar kiln]

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[thick kernel]

I am so confused, I just started so I am stuck at the beginning.

[coral jewel]

can you at least answer the question posed in 2.?

[thick kernel]

I believe its the frustum, correct?

[coral jewel]

correct. next question: What is the formula for the volume of a full cone?

[thick kernel]

How do I do equations in here

[coral jewel]

surround your equations in dollar signs, like $ax^2$

[thick kernel]

$V = 1/3 PiR^2 h$ probably did that wrong

[wraith dagger]

NoodlesMcNoodles

[coral jewel]

what you probably meant is $V = \frac{\pi R^2h}3$

[coral jewel]

what are V, R and h here?

[thick kernel]

Yes

Volume, Radius, and height I assume

[coral jewel]

correct

next question, What technique shall you use to find the formula for the volume of a truncated cone?

[thick kernel]

I'm not sure

[coral jewel]

it does say in the question to watch some video if you are stuck

have you watched it?

[thick kernel]

No

Shoupld I?

[coral jewel]

should you?

[thick kernel]

I guess

gimme a min

Okay so we're thinking about having a smaller cone on top of the truncated cone

[coral jewel]

mhm, does this technique have a name?

[thick kernel]

They did not say

[coral jewel]

personally i would call it as ||complementary counting||, but your video might name it something else

[thick kernel]

All it says is "proportion"

Alright I'll use your definition

Lets move on to 4

How would I make the formula>?

[coral jewel]

well, you already got the essence of it

pretend that you have a cone that is directly connected to the truncated cone's upper base, and so you will have a large cone

[thick kernel]

wait im gonna post this real quick so I have it in front of me

could I just use this?

[coral jewel]

you will use it in some way, yes

[coral jewel]

you can easily calculate the volume of the large cone. but since the small cone wasn't even there in the first place, you need to subtract it from the large cone's volume

indeed, what you are left with is the volume of the truncated cone

suppose that R, r, and h are the radii of the lower base and upper base of the truncated cone and its height respectively, can you derive a formula based on what i said?

[thick kernel]

Im writing it, one sec

That is probably wrong tbh

@coral jewel

[coral jewel]

how exactly did you get this formula?

[thick kernel]

Well you said to add it from the large cone volume right

THe small cone

Actually

wait

nvm

I dont know how to do it

[coral jewel]

this is simpler than you think

everything i just said can be summed up into TRUNCATED CONE = LARGE CONE - SMALL CONE

do you understand where i'm getting at here?

[thick kernel]

We're finding the volume of the entire truncated cone, yes?

And how do I determine how much to take off from the large cone

[coral jewel]

look at the figure here

on the lhs you have your truncated cone, one that you need to find the volume of

think of it as "extending the cone above such that it makes a full completed cone"

by doing that, your completed cone has 2 parts: the truncated part (the one you are trying to find the volume of) and the small part obtained by extending the truncated part

note that the small part is not part of the truncated cone, so your final volume must not include that cone!!

now, do you know the radius of the small cone?

[thick kernel]

No

[coral jewel]

why do you think so?

[thick kernel]

Wait wait wait

We know the volume of the cone but not the truncated cone

[coral jewel]

which cone? there are 2 cones in question here

[thick kernel]

Because I did the formula I got the full one

I think?

4368 pi

for our final

maybe

because 1/3 pi(36)(36) would be 432pi for the smaller one

then the big one would be 4800 after the formula

so I think 4368 would be the final answer?

@coral jewel

sorry for the ping again

[cedar kiln]

@thick kernel Has your question been resolved?

[thick kernel]

.close

[cedar kiln]

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[tawny cape]

This isn’t really help.
But I want to learn more about trigonometry especially more about the unit circle and trigonometric function aswell as identities

[tawny cape]

And there are good websites

Or good pdf books out there

???

Step by step learning

[dire geode]

#book-recommendations

And #geometry-and-trigonometry

[tawny cape]

Anything specific from you??

[void sand]

Khan Academy has an entire section for trigonometry

[tawny cape]

Thanks

[gentle echo]

!done

[cedar kiln]

If you are done with this channel, please mark your problem as solved by typing .close

[tawny cape]

.close

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[slender ginkgo]

1

Mod 3

In which set are we doing this operation

[muted bear]

more like mod erators

<@&268886789983436800> troll

[dire geode]

mod3rators

[honest field]

[muted bear]

what do the three raters think of this joke

[worldly chasm]

@kidgamingmain12 I'm going to close this. Don't abuse the help channels.

[honest field]

mf already left lollll

[worldly chasm]

Oh, already left the server

Coward

[muted bear]

i give it a 1/100, boring, but at least you chose 2+2 over 1+1

[worldly chasm]

.close

[cedar kiln]

Channel closed

Closed by @worldly chasm

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[muted bear]

chickened out

L

what is with the lack of commitment to jokes nowadays

[mighty shuttle]

what is 1+1 ||(In F_2)||

[slender ginkgo]

Blud didn’t answer which set we are in ):

[muted bear]

this one

[dire geode]

In F_2

[slender ginkgo]

💀

[muted bear]

ln(F_2) is clearly ln(F)/ln(2)

[worldly chasm]

Is Set F_3^4?

Or am I missing some other DOF?

[honest field]

F= 96485 C

[cedar kiln]

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[worldly chasm]

@void sand can you explain which condition you find confusing?

Oh, open vs closed.

[void sand]

I was just typing that up in my thread, actually

it's why we want D^(p) to be an open interval containing 0

I suspect that we want 0 just for the group laws that come afterwards

but I'm unsure about the open interval part

[worldly chasm]

This sounds reasonable to me

[void sand]

why not just an open set? why not a half-open interval?

why specifically an open interval

[worldly chasm]

I'm not entirely sure. Let me see if I can find out

[viscid dust]

Yep

[worldly chasm]

Wikipedia mentions this in the section of Vector Flow in Differential Topology:

Let V be a smooth vector field on a smooth manifold M. There is a unique maximal flow D → M whose infinitesimal generator is V. Here D ⊆ R × M is the flow domain. For each p ∈ M the map Dp → M is the unique maximal integral curve of V starting at p.

A global flow is one whose flow domain is all of R × M. Global flows define smooth actions of R on M. A vector field is complete if it generates a global flow. Every smooth vector field on a compact manifold without boundary is complete.

My guess is that it's not a strictly required condition, but they're going to use the restriction later on to assert that if there is a smooth flow in all points of the domain, then we have global flows, which boundaries break.

But it's just a guess.

[void sand]

interesting

hm, that might sate my appetite for an answer for now

thank you @worldly chasm @viscid dust

.close

[cedar kiln]

Channel closed

Closed by @void sand

Use .reopen if this was a mistake.

[worldly chasm]

Also this is cool

https://math.stackexchange.com/questions/1692117/fundamental-theorem-on-flows-lees-book-2nd-edition?r=searchresults

Lee just randomly shows up on a question about his book, confirms an errata and peaces out

[cedar kiln]

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[stark pulsar]

uhh

[stark pulsar]

5(x-y)+4= 7

whats the value of x-y

[exotic furnace]

Think of x-y as one variable

[edgy spade]

let x - y = k (some variable)

[stark pulsar]

mhm

[edgy spade]

so solve for k

[stark pulsar]

my brain got shocked rn

[edgy spade]

since its a linear eqn in one variable (k)

[stark pulsar]

so

x-y is team 1

and k is team 2

but how do i make x and y in the same side

or do i keep flipping

im confused

[edgy spade]

no need for that

[worldly chasm]

Let's try it this way

[edgy spade]

they just need you to find x - y

[stark pulsar]

mhm

[edgy spade]

so you substitute x - y for some variable k

[stark pulsar]

how

[worldly chasm]

5(x-y) + 4 = 7

We've said imagine there's a variable k such that k = x-y

[edgy spade]

you're giving team 1 two different names

[worldly chasm]

Then would you agree we could replace x-y with k anywhere it appears?

[edgy spade]

that is (x - y) and (k)

[stark pulsar]

wait

5(x-y) = 3

[vital jay]

If x-y = k, and 5(x-y)+4 = 7, then 5k+4 = 7

[stark pulsar]

5x-5y=3

then

[worldly chasm]

This is a step in the wrong direction

[stark pulsar]

how

[worldly chasm]

Go back to 5(x-y) = 3

[edgy spade]

yeah

[stark pulsar]

yall

someone said no and someone said yes

[vital jay]

There's different ways to get to the same answer

[edgy spade]

who said no

[worldly chasm]

Read a little more carefully

[edgy spade]

look at which message he said no to

[worldly chasm]

I said no to the distribution

[stark pulsar]

i did

sorry for assuming

[edgy spade]

and look where i said yes to

[vital jay]

What were you going to say here?

[stark pulsar]

ok how do i solve

[edgy spade]

you know what

forget k

[stark pulsar]

isnt it ho

[edgy spade]

find x in terms of y

[worldly chasm]

Well, you want to isolate x-y right?

[edgy spade]

and then find x - y

[stark pulsar]

i got a headache

[edgy spade]

[stark pulsar]

5 (x-y) = 3

[worldly chasm]

How can you change 5(x-y) = 3 into an equation that looks like x-y = some number?

[stark pulsar]

ummm

dont i do like

5 times x

then 5 times -y?

[worldly chasm]

What if instead you divided both sides by something?

[stark pulsar]

hmmm

both by 5?

3/5?

🙂