help-39

[sharp vigil]

have you seen the limit [ \lim_{x \to 0} \frac{\sin x}{x} ] before?

[jolly parrot]

cloud

[static latch]

Wait I searched it up that’s 1

Witchcraft

[sharp vigil]

i would highly recommend you review your one variable limits

[static latch]

thank you.

🙏🙏

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[torpid garnet]

equilateral triangle ABC has sides of 14cm. A circle of a radius of 2 cm it is tangent to sides AB and AC. Find the distance from the circle's centre to side BC in cm

[pearl pond]

Please don't occupy multiple help channels.

[torpid garnet]

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[quartz yoke]

Could someone please help me on how to do this problem?

Not sure where to go from there

[torpid garnet]

this is whawt my ai said i am not a helper To find the minimum perimeter, we need to take the derivative of
𝑃
P with respect to
𝑥
x and set it equal to zero.

Differentiate
𝑃
P with respect to
𝑥
x:

𝑑
𝑃
𝑑
𝑥

2
−
200
𝑥
2
dx
dP
​
=2−
x
2

200
​

Set the derivative equal to zero to find the critical points:

2
−
200
𝑥
2

0
2−
x
2

200
​
=0
Solve for
𝑥
x:

200
𝑥
2

2
x
2

200
​
=2
𝑥
2

100
x
2
=100
𝑥

100

10
x=
100
​
=10
Step 4: Find
𝑦
y Using
𝑥

10
x=10
Substitute
𝑥

10
x=10 back into the area equation
𝑦

100
𝑥
y=
x
100
​
:

𝑦

100
10

10
y=
10
100
​
=10
Conclusion
The dimensions that minimize the amount of fencing are:

𝑥

10
 
m
and
𝑦

10
 
m
x=10mandy=10m
Thus, the rectangular pen should be a 10 m by 10 m square to minimize the fencing required.

[dapper kraken]

let a and b be the sides of the triangle, you know ab=100 and you want to minimize a+b, use am-gm

[quartz yoke]

sorry, what is am-gm?

@dapper kraken

[dapper kraken]

for 2 variables its like this

[quartz yoke]

oh the means

[dapper kraken]

*equality only happens when a=b

[quartz yoke]

oh okay

that makes sense actually

thanks a ton for helping

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[feral leaf]

<@&268886789983436800> gpt response

[torpid garnet]

yessir

[feral leaf]

And here too #help-15 message

[torpid garnet]

yep

i was waiting

but now to tired

to i sleep 😦

Yo my bad i did not read the rules

it wont happen again sir

[pearl pond]

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[velvet wolf]

Can someone please explain the steps to do this?

[pearl pond]

@velvet wolf Has your question been resolved?

[velvet wolf]

<@&286206848099549185>

[pearl pond]

@velvet wolf Has your question been resolved?

[velvet wolf]

<@&286206848099549185>

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@velvet wolf Has your question been resolved?

[warm current]

What does p(0) mean?

[velvet wolf]

polynomial evaluated at x = 0

p(0) = a_0

p(1) = a_0 + a_1x

[warm current]

Oh i see

What is $P_1(\bR)$? I assume a political space but i don’t know of what kind exactly

[jolly parrot]

SWR

[foggy mason]

"political space"

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[unkempt yacht]

what have you tried?

[harsh bobcat]

i tried doing it

i did it by myself and i got d but i put the question in chagtp and i got a

idk whichis right tbh

[urban sun]

chatgpt can be wrong

also u can

[harsh bobcat]

ye i know so thats why im not sure

so i need help please

@vast sluice

[zenith cipher]

You need to use Ax=b
And then find x

Have you made equations in the form of x and y?

[harsh bobcat]

nah

i forgot how to do it and i got a exam tmr

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[pearl pond]

@brisk fiber Has your question been resolved?

[zenith cipher]

!status

[pearl pond]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[midnight haven]

You are not allowed to use this command.

[zenith cipher]

Okay

[cinder flower]

?

[granite tapir]

Just use arc length formula

theta/360 x 2 x pi x R

[west bobcat]

Salut est-ce que quelqu’un pourrait m’aider pour cet exercice ?😅

[granite tapir]

Nos habla espanol

[brisk fiber]

are these answers right?
a. mAB = 26*
b. m<BEC and mBC = 26*
c. mAC = 48*
d. mADB = 26*
e. m<CED and mCD = 154*
f. mDAC = 144*

[brisk fiber]

how about for the first?

[west bobcat]

¿Puedes ayudarme a resolver este ejercicio?

[brisk fiber]

2

[west bobcat]

Ejercicio 1. En una isla tropical hay 2 poblaciones de orquídeas. Estas poblaciones están amenazadas debido a su escaso número. A la primera población sólo le quedan 37 individuos y a la segunda 42. Se prevé un huracán y se estima que la probabilidad de que muera cada orquídea será de 0,95.

1. Calcula la probabilidad de que las 2 poblaciones se extingan cuando pase el huracán (no hay individuos supervivientes).
2. Calcule la probabilidad de que las 2 poblaciones sobrevivan (al menos 1 individuo superviviente en cada población).
3. Calcula la probabilidad de que sólo se extinga una de las dos poblaciones.

Traducción realizada con la versión gratuita del traductor DeepL.com

[brisk fiber]

Este es mi canal para resolver, busque otro canal de ayuda.

[granite tapir]

Too lazy to do it but use the formula for arc length and like add radius and stuff if it's needed

[brisk fiber]

noo like the first image

[granite tapir]

He's speaking french not spanish ;-;

What's m

[brisk fiber]

a. mAB = 26*
b. m<BEC and mBC = 26*
c. mAC = 48*
d. mADB = 26*
e. m<CED and mCD = 154*
f. mDAC = 144*

could you check if these answers are right?

[west bobcat]

🥲

[granite tapir]

whats the lowercase m in front of everything

measure?

[brisk fiber]

yea

measure

[granite tapir]

24/360 x circumference is AB

[brisk fiber]

yea

so are they right\

[granite tapir]

These are probably correct then

[granite tapir]

I guess

[brisk fiber]

could we double check?

a. 26
b. 22
c. 48
d. 334
e. 158
f. 202

[granite tapir]

I'm clueless on how to get the circumference here

But if you have used the correct formula then the answer should be correct too

[brisk fiber]

okay thanks!!

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[torn lodge]

just wanted to check my answer to this question.

x = 3
y = 3
z = 1

[outer shard]

that's right

[torn lodge]

thxx

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[neon jolt]

Hello

[neon jolt]

if we have a differential equation like this : dy/dx=xy^3/sqrt(1+x^2)

and we have ivp y(0)=1

then after solving this eq where should I put the constant C

I mean my final result is sqrt(1+x^2)=-1/2y^2 Then if I put the C in y side I ended up with 3/2. if I put it in x side I ended up with -3/2 result

so which result is correct then?

[north barn]

might have more luck writing this in latex

and writing it longer form in a paragraph

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[neon jolt]

Hello

after solving this eq where should I put the constant C?

my final result is sqrt(1+x^2)=-1/2y^2 Then if I put the C in y side I ended up with 3/2. if I put it in x side I ended up with -3/2 result

So which one is correct I am confused

[lean osprey]

this should be good, you shouldn't have to solve for y, does it tell you to?

[neon jolt]

There's an ivp y(0)=-1 so we have to find the constant

[lean osprey]

and you did tho

c=-3/2, just plug it back into the equation and that's the answer

[neon jolt]

I know.. if I put the C on the other side x side I get different result of the constant 3/2

[lean osprey]

wat

[neon jolt]

After we integrate

We need to add the constant right??

[lean osprey]

yeah, the constant is -3/2

you just found that out

[neon jolt]

Okay hold up

We put that constant in x side right?

If we add the constant on y side and apply ivp we get 3/2

So my question is, Constant C should be always added to x side? Aka dependent side or wut

I am not talking abt my sol

[lean osprey]

are you making an oopsie

[neon jolt]

Wdym?

[lean osprey]

nothing you wrote down is wrong the answer is $\frac{-1}{2y^2}=\sqrt{1+x^2}-3/2$

wat

😭

[neon jolt]

Bruh look

[jolly parrot]

SayMyName

[neon jolt]

So where C should be added after integration

This gives different value for C

[lean osprey]

that's y^2 not y^(-2)

I say be more careful about your writing

[lean osprey]

I guess 3/2 is also a constant that works

but you also have C on the other side

[neon jolt]

Same constant same equation have 2 different values? :/

How can I know then in exam

It'll be multiple choice

[lean osprey]

well they're both right

wait no

[lean osprey]

when you do this, you're multiplying by -1 but not considering the other terms

[neon jolt]

How I multiplied by -1

[lean osprey]

you subtracted C on both sides and changed it to a plus

you can't just do that

[neon jolt]

I didn't substract

We have integration on both sides for the DE so we only put one constant

Can't I just put it on other side instead of choosing right side

[lean osprey]

you're overcomplicating this

[neon jolt]

Anyway thx

[lean osprey]

is your problem solved?

I mean it's the same answer, it's just moved to the other side

f(x)+C=g(x) is the same as f(x)=g(x)-C

[neon jolt]

Actually nope

My teacher said we can put just one constant on any side

But they didn't specify the sign of it they always assumed if we put it in any side it will be positive

So I don't get it tbh

[lean osprey]

actually since both integrals produce a constant, it's something like f(x,y)+C=g(x,y)+D and f(x,y)=g(x,y)+D-C and D-C=E is another constant

or you could do C-D=-E

it's the same procedure

[neon jolt]

I see

Thank you

Have a good day/night

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[limber glacier]

Hello. Can somebody explain to me what branching means in complex analysis?

[pearl pond]

@limber glacier Has your question been resolved?

[limber glacier]

<@&286206848099549185>

[plush bramble]

the best way to understand to start is to see a lot of graphs visually.

start at section 3.3 https://math.mit.edu/~dunkel/Teach/18.04_2019S/slides/L02_ComplexFunctions.pdf

And be sure to understand at least one of the examples in depth. i recommend w = log(z)

[limber glacier]

I'll check this out now. What I've understood so far is this
So if the input is re^(i theta), for the same r, theta and theta + 2npi both represent the same input. If u get different outputs at those points, u must restrict the input to say 0 and 2pi to keep the output single valued?

And the function restricted to that is the branch?

I suppose I do already understand what that document says..

[plush bramble]

yea that's the "many-to-one functions" part of the pdf

[limber glacier]

Mm isn't what I have described a one to many scenario

[plush bramble]

oh yes

sorry yea branching is one to many

[limber glacier]

so we branch to avoid it being one to many

Does z^(1/2) have infinitely many branches?

And does log(z) also have infinitely many branches

[limber glacier]

Yes I understand that it is multiple valued

Depending on how I choose to restrict my input I have different branches?

I was reading a book and it said that z^(m/l) where m and l are coprime, has l branches

I'm not quite sure I understand why that is

[plush bramble]

use $z^a = e^{a \log(z)}$

[jolly parrot]

riemann

[limber glacier]

Yes but my problem is maybe how I can "count" the branches

[limber glacier]

This gives $z^\frac{m}{l}=r^\frac{m}{l}e^\frac{mi\alpha}{l}$, where $z=re^{i\alpha}$

[jolly parrot]

SilverSoldier

[limber glacier]

It's not multiple valued is it?

For any given alpha, I have a single value

[plush bramble]

$e^{m i \alpha / l} = e^{m i \alpha / l + 2 \pi i k},\ k \in \Z$

[jolly parrot]

riemann

[limber glacier]

Okay

[plush bramble]

rearrange the exponent on the right to see multivaluedness of z^(m/l)

[limber glacier]

Rearrange how exactly? e^(a+b) = (e^a)(e^b)?

But that just gives e^(2pik) = 1

[plush bramble]

it's not something you can easily do in one line of math

[limber glacier]

Wait do u mean let alpha=alpha+2pik

[plush bramble]

i think so

[limber glacier]

Okay so $e^\frac{mi(\alpha+2\pi k)}{l}=e^\frac{mi\alpha}{l}\cdot e^\frac{2mk\pi i}{l}$

[jolly parrot]

SilverSoldier

[limber glacier]

Oh..

So a branch still goes from 0 to 2pi

But depending on what k is, each branch looks different. But after k reaches l, it's the same thing repeating, so only l distinct branches

So log (r e^itheta) = ln r + i(theta + 2kpi), a branch is still from 0 to 2pi. For each k I get a branch that looks different to other values of k. So infinitely many branches

Is this correct?

[plush bramble]

the log is correct

the other might need more arguing. or at least another sentence about m being coprime to l

[limber glacier]

Oh okay right

But I've understood it correctly?

[plush bramble]

k reaches l, it's the same thing repeating,
i think m needs to be coprime to l for this to be true

yea

[limber glacier]

Yes it is, that was assumed in the beginning

Oh right then thanks a lot ❤️🫂

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[stiff mauve]

We use chain rule for this right?

[warm current]

product rule

[pearl pond]

@stiff mauve Has your question been resolved?

[pliant valley]

could also multiply it out, then use sum rule and power rule

[stiff mauve]

Got it. when is chain rule used again?

[stiff delta]

compositions

[pliant valley]

function composition

[stiff delta]

sqrt(x^+1) you would use the chain rule here

etc

[stiff mauve]

So if it has an exponent? I'm trying to find an easy for me to remember when to use product rule and when to use chain rule?

[stiff delta]

General rule of thumb:

If you see a function inside a function, use the chain rule like sqrt(x^2+2) or something
If you see a product of two functions like (x^2)(sqrt(x)) use the product rule

you need to practice this more in order to know intuitively

[stiff mauve]

Ohh okokk got it. In sqrt(x^2+2), would

Outer function be sqrt(x^2+2)
and
Inner function be x^2+2 ??

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[haughty timber]

the outer function would just be sqrt(x)

[stiff mauve]

oh yeaa

tyy

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[hot pond]

So I got it to this

Not sure where to go from here

Sorry they’re sideways

[calm wing]

,rccw

[calm wing]

,rccw

[hot pond]

Oh wait

If it’s 1/infinity

Do I just say that it’s equal to 0

So it would just be 1?

Like 1+0

[calm wing]

yes

[hot pond]

And sq of 1 is 1?

[calm wing]

yep

[hot pond]

Ok bet thanks

Have a good day

I got stuck in a L’loop at first

Stupid rule

[pearl pond]

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[woeful shard]

Is integration by parts having 2 different forms that are the same?

I keep seeing

Integral of (f'•g) = f•g - Integral of (f•g')

And

Integral of (f•g') = f•g - Integral of (f'•g)

[tropic saddle]

names dont matter

[pine jay]

The labeling of f and g are not unique and the order in which the multiplication is done could very well be swapped

[tropic saddle]

either way you integrate one of them and differentiate the other

[woeful shard]

I did assume they were commutative but I thought since it's integrals now the same principales couldn't be done just like that

[tropic saddle]

this has nothing to do with integrals

multiplication of functions is commutative because you are just multiplying the outputs which are numbers and multiplication of those is commutative

[woeful shard]

Yep, it makes sense

Thought of 1•lnx and lnx•1 since I was struggling with that prior

Alright, it makes lots of sense, thanks for the explanation you two

Have a good one!

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[dire snow]

using this formula wouldn't it be sin(wt+T) ?

why's it sin(w(t+T))

[pearl pond]

@dire snow Has your question been resolved?

[pearl pond]

@dire snow Has your question been resolved?

[dire snow]

<@&286206848099549185>

[dire snow]

<@&286206848099549185>

[dire snow]

<@&286206848099549185>

[cunning comet]

f(t+T) = f(t) is the definition of a period.
and then use f(x) = sin(wx) and you will get what is written there.

[dire snow]

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[sweet plaza]

Ten points in the plane are given, with no three collinear. Four distinct segments joining pairs of these points are chosen at random, all such segments being equally likely. The probability that some three of the segments form a triangle whose vertices are among the ten given points is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$

ACTUALLY BOUTTA CRASH OUT THIS MFKING PROBLEM WAS SO STUPID
AND IT HAD SUCH A EASY SOLUTION

[jolly parrot]

fpoisonn

[sweet plaza]

<@&286206848099549185> can u guys solve it

[fathom mauve]

I'm not sure if this is right

but the total amount of 3 segment combos is 10 choose 3

or 240

but we need a fourth point

so 45-3=42

so theres 42 choose 4 ways

so 10 choose 3 *42)/ 42 choose 4?

is that it?

[sweet plaza]

ye

ur good at math @fathom mauve this is aime problem #10

[fathom mauve]

wait what

thx

[fathom mauve]

honestly

I find it easier to solve problems like this

in the format What do we have

what information do we need

[fathom mauve]

is that the easy way

or

[sweet plaza]

ig

but its right

[fathom mauve]

oh

If there isn't anything else, please close so others like you can get help!

[pearl pond]

@sweet plaza Has your question been resolved?

[pearl pond]

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[velvet wolf]

Does anyone know what F1 is here?

[pearl pond]

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[pearl pond]

@velvet wolf Has your question been resolved?

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[livid bluff]

Hello, so this is the question, I need help with what you do after the base case. Its not like other induction questions where I can just set P(k) equal to something

[livid bluff]

All I have done so far is showed that the base case holds true, and I am doing the inductive step, where i sub in n for k and i have k(k-1) / 2

[dapper kraken]

well if you add another item, it would be the ammount of subsets that doesent contain the item plus the ones that contain the item right?

the subsets that doeseny contain the item by induction is n(n-1)/2, the subsets that contains the item would be ...

[livid bluff]

hm

sorry am a little

confused

@dapper kraken

[dapper kraken]

what do you not understand?

[livid bluff]

well the last part you said

[dapper kraken]

could you think of what it would be?

[livid bluff]

wouldnt it be like

k+1

assuming i am subbing k for n

[dapper kraken]

not quite

think about this, you have k+1 total items, you already picked the last one, so how many are left that you can pick?

[livid bluff]

ohh

k?

[dapper kraken]

yes

[livid bluff]

ok but how do I like

use that

[dapper kraken]

now if you add k(k-1)/2+k?

[livid bluff]

one sec ima think abt this

ohhh

wait i got it

ok all I do is add k to the original

then I show that it equals to k + 1

i think

[dapper kraken]

yes

[livid bluff]

ok thank you so much

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[stiff mauve]

I need to keep simplifying but I don't know where to go from here. Do I give the numerator section a denominator of (e^x+1)?

[inland ivy]

Your answer already matches one of the answers

You just need to cancel an x

[stiff mauve]

Frr??

Could someone draw where the x's in my work cancel out 😭

[stiff mauve]

OH, didn't know you could just do thattt

tytyyy

[compact ridge]

yeah just factor out an x from the numerator

that's how

[stiff mauve]

OHH

[compact ridge]

you need both xs to cancel in the numerator

[stiff mauve]

got it gott itt tyvmmm

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[midnight haven]

i got the first part down im just looking for a bump in the right direction on the second part.

[pearl pond]

@midnight haven Has your question been resolved?

[midnight haven]

<@&286206848099549185>

[cursive pier]

saturn my broski you already did the hard part, part 2 is just using the length from 1 And the angle to get the height of the triangles, both should give the same height if youve cheffed it up right

[midnight haven]

hmm can you elaborate a bit more if you dont mind?

would I just do like

445tan(44.5) to get the height?

<@&286206848099549185>

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[midnight haven]

For 1

[midnight haven]

This is how we write it
Right?

[gilded loom]

Yep

[rough forge]

Yep

[midnight haven]

Okay ty

.close

[gilded loom]

Np

[pearl pond]

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[midnight haven]

.reopen

[pearl pond]

✅

[midnight haven]

,rccw

[midnight haven]

4the line

"this can be visualised by thinking...."

I couldn't do jt

Could someone help

Or send a video explaining how to visualise average value

[cursive wraith]

if you have a tank of infinite height

and let each point (x,y) have a particular water height f(x,y) at the beginning

the total volume of water in the tank is just integral(f) over the base of your tank, right?

[midnight haven]

sorry was afk

[midnight haven]

total volume = integral f(x,y)dA yes

over the base means that,right

[cursive wraith]

ok

[cursive wraith]

yeah the base is R

so

now, let's just think about what's gonna happen physically

you have a body of water

with possibly some water heights higher than others

so if you let time run through

the water itself is gonna average out its height

[midnight haven]

yeah

[cursive wraith]

as long as there is some points where the height is bigger, the water's gonna flow out of that point

and vice versa where height is smaller

[sharp vigil]

https://www.desmos.com/3d/eqbagq7bvq

[cursive wraith]

so let time run through

and after waiting for a (maybe very long) time

the height of the water is the same at each point of the tank, call it H

what's the volume of water in the tank?

[cursive wraith]

also mb the base is called R

[midnight haven]

area times height ?

[cursive wraith]

area of the base times height yes

[midnight haven]

area of base *

yeah

[cursive wraith]

so area(R)*H

but the volume of water hasn't changed since the experiment started

[midnight haven]

oh i realized i made 2 mistakes until now when i did it on my own

[midnight haven]

yes

[cursive wraith]

so this is still the volume of water in the tank

[midnight haven]

so

[cursive wraith]

H = 1/area(R) int(f(x,y)dA)

[midnight haven]

where did average come over here

nvm

nvm

sheesh

this chapter is way better than partial derivatives

thanks

.close

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[midnight haven]

.reopen

[pearl pond]

✅

[midnight haven]

,rccw

[midnight haven]

24

@sharp vigil

the method you told me to find the limits

,rccw

[midnight haven]

so when going from left to right the first curve we approach will be the lower bound

but here

[sharp vigil]

one thing you can do is take advantage of the symmetry of the shape

[midnight haven]

yeah i did

still stuck

till here theres no problem

[sharp vigil]

the shape is identical in all 4 quadrants, so you only need to integrate it in one quadrant (and take 4 times your result)

[midnight haven]

there is no curve here

its creepy

[midnight haven]

i did that

[sharp vigil]

there is the y-axis there

[midnight haven]

OHHHHHH

😂 dum me

[wild star]

Guys its me

[midnight haven]

oh why take the pain of adding 3 doule itegrals

i can just substract

Shit +

,rccw

Is this right ?

[sharp vigil]

why is 0 your upper bound for x?

[midnight haven]

Oh it should be the reverse

First one I hit is y axis

B

Mb

Is this correct?

[sharp vigil]

seems fine

[midnight haven]

okay

tyy

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[midnight haven]

.reopen

[pearl pond]

✅

[midnight haven]

25

one image is 19MB

density

total population

units dont match

https://tenor.com/view/confused-anime-crazy-dizzy-gif-16634430

[sharp vigil]

population density = population per unit area

[midnight haven]

oh

i was taking density as mass / volume and wondering why the question was making no sense

1 thing

[midnight haven]

even thought the value of x is negative i should still be going from left to right while finding the bounds right ?

@sharp vigil last one

[sharp vigil]

density can be with respect to length, area, or volume depending on the context

yes, the bounds always go left to right

[midnight haven]

alright

Thanks a lot!

.close

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[tiny escarp]

Is there any way to find the cosx and sinx if I know the sum of cosx+sinx

[signal flicker]

you need more context

or more information

you can't do anything if you just know the sum of cos(x) and sin(x)

actually

you can solve it

do you know the identity cos^2(x) + sin^2(x) = 1?

[tiny escarp]

Sure

[signal flicker]

well you can isolate either the sin(x) or cos(x)

and you get cos^2(x) = 1 - sin^2(x)

so cos(x) = sqrt(1-sin^2(x))

and you know that cos(x) + sin(x) = some value let's call it y

[tiny escarp]

Okay then I can write sqrt(1-sin^2(x)) instead of cosx and solve second degree

[signal flicker]

yeah

[tiny escarp]

OK I will try again

[signal flicker]

sqrt(1-sin^2(x)) + sin(x) = y

[tiny escarp]

But that method somehow doesn’t work

Or I do calculation mistake

[signal flicker]

what is the sum you have?

send me your calculations

sorry I'm not really a helper in this channel but this problem doesn't seem bad I should be able to help

[tiny escarp]

Wait one minute I think I found my mistake

I will try again

If can’t

I ll send you

[signal flicker]

ok

[tiny escarp]

No I couldn’t

[signal flicker]

wait how did you get -2sqrt2 sin(x) = 3/2

oh is that the equation you started with?

so we have sqrt(1-sin^2(x)) - 2sqrt(2)sin(x) = 3/2

you can solve it by doing u-substitution

honestly that seems pretty nasty though

[pearl pond]

@tiny escarp Has your question been resolved?

[tiny escarp]

Yeah I thought the same

Thanks btw

[pearl pond]

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[bitter herald]

how did they arrive at (3.105)?

[pearl pond]

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[late sun]

How would I go about integrating $e^{4x} \cos 5x \sin 2 \sqrt{2} x$ I tried to turn sin into cos, but it makes it look worse

[jolly parrot]

lewis_f04

[honest spear]

product to sum trig identity?

[desert swallow]

tbc where exactly are the brackets in that. is it $e^{4x} \cos \left( 5x \right) \sin \left( 2 \sqrt{2} x \right)$?

[jolly parrot]

meow mrrrrrrrrow mrrrp mrrrow

[desert swallow]

my instinct is integration by parts w/ e^4x as u and cos(...)sin(...) as v

[honest spear]

try this

@late sun

[late sun]

That’s correct

[late sun]

I’ve never seen that identity but I’ll try that thanks

[late sun]

That worked thanks a lot

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[jolly parrot]

Halex

[pearl pond]

@midnight reef Has your question been resolved?

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[tranquil root]

$$20x-(x-10)^2=$$
$$20x-(x^2-20x+100)=$$
I know how to simplify this expression, but the minus sign behind the parenthesis just doesn't make any sense to me.
I mean, minus is to subtract, so why do we use the distributive property (aka multiplication), while assuming this expression is the same??:
$$20x -1\times(x^2-20+100)$$
although it's subtracting, it just doesn't make sense.

[jolly parrot]

𝕿𝖆𝖘𝖐

[tropic saddle]

its more 20x+(-1)*(x^2-20x+100)

if you want to think of it as distributing a -1

[tranquil root]

ok, but it's not my point

[tropic saddle]

then what is your point

[tranquil root]

my point is, that the operation is subtraction not multiplication

[tropic saddle]

-x=-1*x?

[tranquil root]

yeah

[tropic saddle]

you can just do all of this in terms of addition and subtraction if you want

-x means the thing that when added to x gives 0

so -(x^2-20x+100) is the thing that when added to x^2-20x+100 gives zero

which turns out to be -x^2+20x-100

[tranquil root]

ok
still don't get it, how is it that we can just replace subtraction with multiplication?

[tropic saddle]

because of the distributive law we have that x+(-1)*x=1*x+(-1)*x=(1+(-1))*x=0*x=0, so (-1)*x is the thing that when added to x gives zero. aka -x=(-1)*x

[tranquil root]

hmmm

[pearl pond]

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[wind sand]

Hello, I have a question regarding p values. If I set alpha to 0.05 and if my null hypothesis is H0: "X and Y are independant". If the p value is lower than 0.05, it is very unlikely that X and Y are really independant, so I can reject H0. Now, it seems intuitive to say that if the p value is higher than 0.95, it is very likely that the null hypothesis is true <=> X and Y are independant, but I have never seen this in the material I read. Can you draw conclusions if the p value is higher than 1 - alpha ? And if no, could you explain why that is

[humble lintel]

If you already have the p-value and the alpha, making the conclusion that something is likely or unlikely to occur is much less precise than what you originally had

[pearl pond]

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[pearl pond]

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[somber heron]

HELP! Idk how to solve for the angle. No matter what method(law of sin,cos or angle between 2 vectors method )i use i kept getting the same answer, 27.24 degree. But its still different from the solution provided. What did i miss here or is it simply the website's mistake again?

[somber heron]

<@&286206848099549185>

[hoary hull]

Well we know what ways west and southwest is and can calculate points at each end. It will be between west and southwest, can you do the rest? X E.

[clear moon]

the theta you drew is also very misleading

[somber heron]

pink and red theta?

they are both referring to the same angle tho

[somber heron]

thats what i did tho, and i got the distance correct, but not the direction

[hoary hull]

Did you reverse which way total of vector went?

[somber heron]

oh tan inverse?

i just did, and i got the correct answer...

[hoary hull]

Happy to help. X E.

[somber heron]

but i dont get why law of cos and sin didnt work

[chilly sand]

i dont quite understand the problem

[clear moon]

the theta in your diagram isn't what you're measuring as the degrees north from east

you'll want to find the angle from red and purple, since that's with respect to your coordinate system

maybe this is clearer

you could also use your current work and have 90 - 45 - 27.24

if you make a rectangle from the origin to the end of the blue vector

[somber heron]

i see, so i was looking at the wrong angle

[clear moon]

yep!

[somber heron]

i actually did think of that, but i think my brained bugged earlier i did -90 instead lol

[clear moon]

oops haha

[somber heron]

Its mid night now i should go sleep now

[clear moon]

sleep is important for learning!!

[somber heron]

thanks for answering my question

[clear moon]

ofc!

[somber heron]

.close

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[thorny shore]

If I’m rolling a 6 sided dice. My mean is 5/2. My variance is 35/12. But what is my variance really saying. Variance is” a measure of how spread out a set of numbers are from their average value”. Well none of my numbers, 1-6. Are 35/12 away from the mean or from eachother. And they aren’t on average 35/12 away from each other or the mean. So what is the variance really saying in this case?

[ivory swallow]

variance is the difference between the 'mean of squares' and the 'square of mean'

note that this is not in the same 'units' as the original

for instance, if each number on the die corresponded to a length in meters, then the variance would be in meters squared

take the square root of the variance to obtain the standard deviation, which is in the same units

and that gives you a useful measure of how far each point is from the mean

[pearl pond]

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[pearl pond]

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[neat panther]

In the end I guess I have a problem with final ans?

[neat panther]

It's the opposite of what I wrote

I had -1/64 * ln2 - 15/256

[tacit rain]

what is the question

oh i aint familiar with that

[midnight haven]

[neat panther]

Nvm I'm stupid

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[pearl pond]

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[lyric stratus]

So there's this streamer that makes weekly raffles which you enter with points that you earn by watching his stream, now there's 5 different raffles, the first one has 5 130$ prizes, about 12000 participants, and each ticket is 1000 points, the second one has 10 60$ prizes, about 16000 participants, and each ticket is 500 points, 3rd 20x35$, 29000 participants, 300 points, 4th 30x22$, 41000 participants, 200 points, 5th 40x10$, 47000 participants, 100 points.
Now if the ticket entry was a 1$ I could calculate the EV of each raffle aswell as know how many tickets I should buy to maximize that EV, but since its points I get really lost, I haven't studied math in years so even worse.

[jolly parrot]

⭐Kyrosma⭐

[plush bramble]

unfortunate

use backticks ``` around your entire message

[lyric stratus]

So there's this streamer that makes weekly raffles which you enter with points that you earn by watching his stream, now there's 5 different raffles, the first one has 5 130$ prizes, about 12000 participants, and each ticket is 1000 points, the second one has 10 60$ prizes, about 16000 participants, and each ticket is 500 points, 3rd 20x35$, 29000 participants, 300 points, 4th 30x22$, 41000 participants, 200 points, 5th 40x10$, 47000 participants, 100 points.
Now if the ticket entry was a 1$ I could calculate the EV of each raffle aswell as know how many tickets I should buy to maximize that EV, but since its points I get really lost, I haven't 

[limber oasis]

you can compute the average $/point of each raffle
it's just (total money in the raffle = nb of prizes * cash prize) / (number of participants * cost of entry)
And maximize that
Though since this is a raffle, the most likely outcome is that you don't see any of that money. Even the most likely one is merely 0.1% chance of 10$

[jade flicker]

I was going to say this, you're assuming that there's enough people that each individual entry doesn't change the probability that the next one wins - might be worth stating if this is for submission but probably doesn't matter

[limber oasis]

yeah cause you're not buying a thousand tickets
And even if you are, it's not a big statistical change

[lyric stratus]

forgot to mention that you can also bet those points on league of legends games and accumulate, I now have around 2 million so there's chances to win, i'll try with this thank you

[limber oasis]

yeah that's about 100-150$ worth depending on the raffle and your luck (could be much more or much less)
Quite the grind surely

[lyric stratus]

yeah thats I what I got by calculating my chances of winning each raffle, my doubt was wether I should max out on a single raffle or distribute the points across all of them, aswell as at what point do I get diminish returns with each ticket. I found this online about the best number of tickets to buy in a raffle so if I multiple the average $/point in each raffle by that third x I should get the optimal number of tickets per raffle right?

x is the number of tickets

[limber oasis]

the last raffle is just much better than the others
If you decide to dump all your points into 20k tickets then yeah suddenly there's 70k tickets it's not that good
But if you do it over, say, a few months, spreading it over 10 or 20 raffles (it's weekly right?), then the effect is negligible and you maximize your income, because this will be much better than how much worse the 2nd to last raffle is compared to the last

[limber oasis]

cause if you did the computation, I think the last one comes as as way better

[lyric stratus]

yeah but the thing is I'm expecting to make around a million points each week

[limber oasis]

holy

how much time do you spend on this?

[limber oasis]

did you mean participants, or tickets bought?

[lyric stratus]

tickets bought

[limber oasis]

yeah so there's not actually many active people if you can just get that many tickets

[lyric stratus]

not much, you get 100 points per 10 minutes watched, but you can scale that a lot higher with bets

[limber oasis]

which means it's weird the guy can just pull 12k/month on a lottery that actually has like, 200 active people?

[lyric stratus]

he streams slots and has an average of 3k viewers

so its all sponsored by the casino

[pearl pond]

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[midnight haven]

I just need someone to explain conditional probability to me pls

[hushed sky]

yeah for sure, what're you not understanding specifically?

or is it just conditional probability in general

[midnight haven]

in general

@hushed sky you free?

[hushed sky]

@midnight haven yeah sorry lol

[midnight haven]

ok

[hushed sky]

so basically you've been working with normal probability (im assuming)

[midnight haven]

yes

[hushed sky]

conditional just means if this happens, the probability of the other event will change

[midnight haven]

ok

[hushed sky]

so here's an example on unconditional vs. conditional

unconditional: i have 10 marbles in a bag, 5 red and 5 blue. i pick one at random, it turns out being red. i put it back in the bag. what is my probability of choosing another red?

[midnight haven]

is it P(A)

[hushed sky]

yeah, but whats the probability

like give me a number lol

[midnight haven]

oh

hold on

fraction or deci?

[hushed sky]

doesnt matter

[midnight haven]

0.5?

[hushed sky]

yep

and that's kind of obvious right

[midnight haven]

a bit

[hushed sky]

because we have 5 red marbles out of 10 marbles

so it's just 5/10 = 0.5

[midnight haven]

yep

[hushed sky]

ok now for conditional

conditional: i have 10 marbles in a bag, 5 red and 5 blue. i pick one at random, and it turns out being red. i don't put it back in the bag. what is my probability of picking out another red?

[midnight haven]

i have no relative idea, im guessing 4/9 from what i can figure out

[hushed sky]

and guess what

you're right lol

[midnight haven]

wow

how

i need to know

[hushed sky]

our number of red marbles decreases as well as the total number of marbles in the bag

[midnight haven]

oh

[hushed sky]

so now we have 4 marbles

and out of 9 marbles

so it's 4/9

this is what conditional probability is

[midnight haven]

ok

[hushed sky]

one event happens, and then we find the probability of the 2nd event happening given that the first one changes the 2nd one

a bit wordy

if my explanation makes no sense, just put this prompt into chatgpt:

[midnight haven]

does it always decrease by 1

[hushed sky]

explain conditional probability and unconditional probability like im 10

[midnight haven]

u make sense dw

[hushed sky]

okok

[hushed sky]

hmm no, because let's say we took 2 marbles out instead

and they're both red marbles.

now our probability is 3/8 for picking another red marble

[midnight haven]

let me think pls

[hushed sky]

👍 take ur t ime

time*

[midnight haven]

ok

[midnight haven]

i fully understand

now

[hushed sky]

awesome

what grade level are you in?

[midnight haven]

what do you mean?

[hushed sky]

like what grade are you

[midnight haven]

im british so in american terms im sophomore ill say?

[hushed sky]

okay

so i think this is the formula you'll learn in school then

[midnight haven]

correct but

the upside down u represents?

[hushed sky]

it means intersect

probability of A and B

which means to multiply their probabilities

[midnight haven]

Ok

Hey, are you still free?

[hushed sky]

yeah

[midnight haven]

Ok

What i'm about to ask is a bit harder

Please tell me the steps to working this out

A box contains 5 red discs and 3 green discs. One disc is taken at random and its colour noted before being replaced. A second disc is then taken. Find the probability that both discs are the same colour.

[hushed sky]

so first is to look at what are the possible outcomes

we either get 2 red discs or 2 green discs, right?

[midnight haven]

yes

[hushed sky]

so whats the probability of getting 1 red disc first

[midnight haven]

is it 0.2

[hushed sky]

5/8

is what im getting

[midnight haven]

i mean 0.125

[hushed sky]

why 0.125?

[midnight haven]

because i added 5 and 3

so u got 0.625

[hushed sky]

yesd

[midnight haven]

how

[hushed sky]

because the probability of getting a red is 5/8

there are 5 reds and 8 total discs

[midnight haven]

oh i understand

nextly?

[hushed sky]

now we replaced the disc (we put it back in the bag)

what's the probability of pulling out ANOTHER red?

[midnight haven]

0.5???

[hushed sky]

how are u getting 0.5?

[midnight haven]

4 / 8

[hushed sky]

no because we replaced it remember

we put the red disc back in, so our total red is 5 again

[midnight haven]

oh i assumed it was taken out

ok

[hushed sky]

it's 5/8 again

so that means to pull out 2 reds, our total probability is (5/8) * (5/8)

but we can also get 2 greens, so that would be (3/8) * (3/8)

since we can get either of them, we add their probabilities

[midnight haven]

ok

state what to add pls

[hushed sky]

bro 😭

okay i get that it might be a bit confusing, but u need to try to learn it urself

like it's pre obvious that u're just using this for an assignment

anyways ur final probability is 0.53125

im gonna tell u now that u have to show some interest in studying on ur own rather than just grabbing answers

for ur own benefit, just use chatgpt to help explain it to u

[midnight haven]

oh no, im really not.

[midnight haven]

this is exactly what im doing

i just wrote my math mocks and im not so confident in it

the question i gave u is an example from my revision guide thats it

i wanted to know how you would apply the formula to it

probability is my weakest area, im trying to get better

do you refuse to help me further?

[errant wadi]

I'm gonna give some general probability tips if you're just starting out

[midnight haven]

ok

[errant wadi]

there's really only three things you need to know to solve probability question like this

1. Probability of something happening = Number of ways that thing can happen / Total cases
2. Probability of A and B happening = Probability of A happening * Probability of B happening (as long as A and B are independent, so they don't influence each other)
3. Probability of A or B happening = Probability of A happening + Probability of B happening - Probability of A and B happening

idk how understandable that is but I can break it down more if you want

[midnight haven]

let me read it

[midnight haven]

hopefully u can see it clearly as a piece of proof/evidence

ok back to reading

AND, OR for 1.) and 2.) ?

I mean 2.) and 3.)

[errant wadi]

yeah

[midnight haven]

Ok

Im not sure if I get it or not

Maybe I do

[errant wadi]

for instance, with this problem:
the probability of both being the same color = probability of them both being red or both being green.
probability of them both being red or both being green = probability of them both being red + probability of them both being green - probability of them both being red and green

probability of them both being red and green = 0 obviously since that's impossible
probability of them both being green = probability of one being green * probability of one being green = 3/8*3/8 = 9/64
probability of them both being red = probability of one being red * probability of one being red = 5/8*5/8 = 25/64

final answer: 9/64+25/64=34/64

when you get better with probability stuff you end up using those three rules naturally without thinking about it

[midnight haven]

Ok

[errant wadi]

Hope that helps a little, I'm can be iffy at explaining things sometimes

[midnight haven]

I'm still reading, I'll let you know

A box contains 5 red discs and 3 green discs. One disc is taken at random and its colour noted before being replaced. A second disc is then taken. Find the probability that both discs are the same colour.
For my personal clarification

[midnight haven]

Can you pls break it down more?

[errant wadi]

ok

I want to find what part I need to break down

probability of them both being green = probability of one being green * probability of one being green = 3/8*3/8 = 9/64
probability of them both being red = probability of one being red * probability of one being red = 5/8*5/8 = 25/64

do these lines make sense?

[midnight haven]

Ok, it does make sense, but the problem is I don't want to get it mixed up with other types of probability.

And I tend to do this a lot

[errant wadi]

what do you mean?

[midnight haven]

Wait..

AND / OR events

'At least' events

[errant wadi]

if you follow these rules you'll never be wrong with AND/OR events

[errant wadi]

can you give an example of what you mean by an 'at least' event?

[midnight haven]

I roll a 3 fair 6 sided dice. Find the probability that I roll at least 1 six.

[errant wadi]

Ah

at least events like that are really just OR events in disguise

[midnight haven]

Ok

[errant wadi]

I roll a 3 fair 6 sided dice. Find the probability that I roll at least 1 six.
is the same as
I roll a 3 fair 6 sided dice. Find the probability that I roll 1 six, 2 sixes, or 3 sixes

[midnight haven]

Nice

[errant wadi]

but there's actually a better way to do this question

[midnight haven]

Tell me

[errant wadi]

What's the probability you don't roll at least 1 six

[midnight haven]

Forgive me if I get it wrong

5/6?

[errant wadi]

that's if you roll 1 die

what if you roll 3?

[midnight haven]

Oh 3?

9? (3 x 3)

9 / idk

[errant wadi]

The probability you don't roll a six on the first roll is 5/6.
The probability you don't roll a six on the second roll is 5/6.
The probability you don't roll a six on the third roll is 5/6.

the probability you don't roll a six on any of them is 5/6*5/6*5/6

[midnight haven]

Oh.

[errant wadi]

it's an AND statement

[midnight haven]

125/216, then?

[errant wadi]

Yes

So if that's the probability you don't roll at least 1 six, what's the probability that you do roll at least 1 six?

[midnight haven]

In one die or 3 dice?

[errant wadi]

3

[midnight haven]

Definitely 91/216

[errant wadi]

Yep!

[midnight haven]

Pls, explain what you would do after finding the probability of not rolling one six

And justify why

[errant wadi]

You just take 1-125/216, just like you did

[midnight haven]

Oh. Ok

Let me try to absorb this now.

[midnight haven]

When you say this, what exactly do you mean?

[errant wadi]

Well when you roll 3 dice, either you roll at least 1 six, or you don't.

so the probability that either one happens is 1

we know that the probability that you roll no sixes is 125/216

[midnight haven]

Yes

[errant wadi]

so the probability you roll at least 1 six is 1-125/216

[midnight haven]

Ok, thanks.

I either watch TV or read before bed. The probability I watch TV is 0.3. If I read, the probability I am tired the next day is 0.8. What is the probability that I read and am not tired the next day?

How do you break down this question?

And apply the AND rule

[errant wadi]

Well what's the proabbility that you read?

[midnight haven]

0.7 is it?

[errant wadi]

yep

If you read, what's the probability that you're tired the next day?

[midnight haven]

0.8

[errant wadi]

What's the probbaility you aren't tired if you read?

[midnight haven]

Wait, firstly, should 1 - 0.8 = 0.2?

If so what why and what next

[errant wadi]

yeah that's right

[errant wadi]

same type of logic as up here

you're either tired or not

so the probability of either one is 1

[errant wadi]

So we want both events to happen, and with AND events you multiply probabilities.

[midnight haven]

I think this is JUST what I needed.

0.7 x 0.2 = 0.14

[errant wadi]

Yep

[midnight haven]

So, to fortify my knowledge, can you select any examples we just went through and change what the question asks according to AND, and OR?

Select one question of your choice but word it in 2 different manners

I want to compare the differences

[errant wadi]

A box contains 5 red discs and 3 green discs. One disc is taken at random and its colour noted before being replaced. A second disc is then taken. Find the probability that both discs are the same colour.

A box contains 5 red discs and 3 green discs. One disc is taken at random and its colour noted before being replaced. A second disc is then taken. Find the probability that (First disc is red and second disk is red) or (First disc is green and second disk is green)

[midnight haven]

First AND second OR?

[errant wadi]

in this case yes

[midnight haven]

Ok

Any other possible way it could be changed?

[errant wadi]

Not any obvious ways that I can see

[midnight haven]

Ok

For the first question what is the most important information that you need to start solving?

And probably the only information anyone could need

So take out the parts of the sentence that are irrelevant and re - type it

[pearl pond]

@midnight haven Has your question been resolved?

[pearl pond]

@midnight haven Has your question been resolved?

[midnight haven]

.close

[pearl pond]

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[pearl pond]

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[forest sun]

hi im trying to use wolfram Mathematica to find the 20th decimal but i don't get where i'm wrong?

[buoyant linden]

use a capital e

E is 2.718281828459045….

e is just a variable

[pearl pond]

@forest sun Has your question been resolved?

[forest sun]

ah i see

thank you i'll try that

.solved

[pearl pond]

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[pearl pond]

Available help channel!

Send your question here to claim the channel.

Remember:
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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.

[wild temple]

Unit#4: Factorization and Algebraic Manipulation

Can someone explain this to me?

[mint wave]

what part are you stuck on

[wild temple]

I am stuck on

3rd part

I mean the 3rd part looks confusing because the 4th is so long.

[midnight haven]

in the third part, we are essentially completing the square

[wild temple]

What does completing square mean? Can you simplify?

[midnight haven]

yup

so basically

when we have an equation of the form

$$a^2 + 2ab$$

[jolly parrot]

Edmund Cloudsley

[midnight haven]

and we wish to factorise the equation,

we add $-b^2$ and $+b^2$