help-42

[humble steppe]

for this one

@bold sedge what technique is best to solve this integral

u sub?

nvm

its not u sub

[bold sedge]

Yes. Substitute 1-y=u

[humble steppe]

oh..

[bold sedge]

You can do it using u sub easily

[humble steppe]

not by doing partial fraction decomp?

[bold sedge]

What will you decompose it into.

The denominator is linear

[humble steppe]

idk but with u sub 1+y wills tay the same

[bold sedge]

Yes

[bold sedge]

Thats why do this.

[humble steppe]

i just get -1(1+y)^2 du

[bold sedge]

Wrong

Do the substitution correctly

[humble steppe]

-(1+y)^2/u du

[bold sedge]

You should replace 1+y in terms of u.

[humble steppe]

idk how to

[bold sedge]

If 1-y=u then
1-u=y
so 2-u=1+y

[humble steppe]

if u =1-y then 1+y is u+2y

oh

then whats wrong with my reasoning?

[bold sedge]

Because y is still there

You have to replace all instances of y with u.

[humble steppe]

okay i get (-2+u)/u

[bold sedge]

Again wrong

[humble steppe]

bruh

[potent lotus]

bathroom mug

[humble steppe]

i forgot tthe 62

^2

but this integral is still hard to do

[bold sedge]

No

$-\int \frac{(2-u)^2}{u} du =-\int \frac{4+u^2-4u}{u}=-\int (u-4+4/u) du$

[potent lotus]

bathroom mug

[bold sedge]

I hope you can do on your own now.

[humble steppe]

oh i see

thank you

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[calm coral]

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[next wren]

forgot the measure once :P

[bold sedge]

Yeah second equality, i forgot the du.

It is usually called the differential.

[next wren]

lebesgue measure?

[bold sedge]

Unless you are doing measure theory

[calm coral]

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[safe matrix]

could you send a paper about laplace pressure of uneven shapes?
(my question timed out twice so ill ping <@&286206848099549185> )

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[coarse minnow]

6 persons enter an elevator and pick - independently and with uniform probability - one of 8 floors.
Calculate the expected number of floors on which some number of people (>0) leave the elevator.

Note: Consider a random-variable that is 1 exactly when a person gets off on this floor and 0 otherwise.
First find the probability with which this variable is 0, then its expected value.
With this expected value you can easily calcaulte the result we're looking for.

[coarse minnow]

I'm not sure where to even begin with this one

The random-variable they're looking for for each floor is 0 exactly when all 6 people choose to not get off at one (out of 8) given floors so with probability (7/8)^6 ?

so it's 1 with a probability of 1-(7/8)^6

so since it's binary that's the expected value here?

now how does that help with finding the expected number of floors?

[calm coral]

@coarse minnow Has your question been resolved?

[civic dirge]

We can make another random variable that represents the number of floors people choose to get off the elevator

Xt = X1 + X2 + ... X8

The actual distribution of this random variable is probably very difficult to get. However, the expectation is easy.

@coarse minnow

[coarse minnow]

is the expected value the sum of the individual expected values?

[civic dirge]

Exactly

[coarse minnow]

so approximately 4.41?

$\sum_{i=1}^8 1-(\frac{7}{8})^6 = 8(1-(\frac{7}{8})^6) = \frac{144495}{32768} \approx 4.41$

[potent lotus]

Bob Goldham

[coarse minnow]

this?

[civic dirge]

Yeah it makes sense to me

[coarse minnow]

probabilities making intuitive sense usually means that they're wrong..

[civic dirge]

It seems easier than it should be, haha. Expectation is easy like that. I have the urge to code this and see the result

[coarse minnow]

I just did that

#!/usr/bin/env python3
import random

LEVELS = [1,2,3,4,5,6,7,8]

def run_once(n_person: int = 6)-> int:
    chosen_levels = set()
    for i in range(n_person):
        chosen_levels.add(random.choice(LEVELS))
    return len(chosen_levels)

def mean(data):
    n = 0
    mean = 0.0

    for x in data:
        n += 1
        mean += (x-mean)/n
    if n < 1:
        return float('nan')
    return mean

def main():
    results = []
    for i in range(1000000):
        results.append(run_once())
    print(mean(results))

if __name__ == "__main__":
    main()

This does consistently produce values around 4.41, so I'll assume the solution is correct

thank you @civic dirge !

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[sharp wasp]

The different birthdays of 𝑛 people must be at least 3 days apart from each other. Determine approximately
𝑛 such that the probability of this happening is 50%

[sharp wasp]

is this right??

[mortal orbit]

why 365 - 3(n-1)?

[sharp wasp]

i dont know i just came up with that

[mortal orbit]

yeah it's not correct

[sharp wasp]

how would u do it

maybe 365-3n/365^n=0.5 starting from n=0?

[mortal orbit]

no that's not it either...

take n = 2

[sharp wasp]

why n=2?

[mortal orbit]

for example

so that you see what's wrong

the probability that two birthdays are at least 3 days apart from each other is 365*358/365^2, can you see why?

[sharp wasp]

shouldnt be 365*(365-3)/365^2?

[mortal orbit]

no

[sharp wasp]

we remove 3 days for both of them

[mortal orbit]

no that's still not it

suppose the first birthday is chosen

[sharp wasp]

hahaha im good at this

[mortal orbit]

which birthdates are LESS than 3 days apart?

[sharp wasp]

this is 365/365 right?

[mortal orbit]

yes

say we picked june 10th

[sharp wasp]

yes

[mortal orbit]

it doesn't matter but it's for representation

which birthdates are NOT at least 3 days apart from june 10th?

[sharp wasp]

7-8-9, 11-12-13

[mortal orbit]

you're missing something

[sharp wasp]

7-8-9, 11-12-13 this is the 3 days that 10 is apart from

[mortal orbit]

ok, so many things wrong with that sentence

so

june 7th, yes it's not at least 3 days apart

because |7-10| = 3, is not greater than 3

same reasoning for 8,9, 11,...

but you missed one

[sharp wasp]

i cant figure it out

[mortal orbit]

don't look far

the days that are not at least 3 days apart from june 10th are not further "before" june 7th and not further "after" june 13th either

but you still missed one

7,8,9,11,12,13

are those the ONLY days that are 3 days apart or less from june 10th?

[sharp wasp]

yes thats right

so for the 10 june we remove this days 7,8,9,11,12,13

[mortal orbit]

yes but which day do we remove as well?

[sharp wasp]

10?

[mortal orbit]

YES!

[sharp wasp]

YESSS

[mortal orbit]

finally

10 is not at least 3 days apart from 10

ok

so 7 days

we remove 7 days

so 358

[sharp wasp]

so 365*(365-7)/365^n

7n

[mortal orbit]

?

no

[sharp wasp]

we start from n=2

??

[mortal orbit]

ok you're bad at inferring formulas

n=1 was what

[sharp wasp]

yes

[mortal orbit]

what's the probability when n=1

that all birthdays are 3 days apart or more from each other

[sharp wasp]

365/365?

[mortal orbit]

yes

n = 1 :

365/365

n= 2:

[sharp wasp]

358/365

[mortal orbit]

365(365-7)/365^2

[sharp wasp]

yes yes

[mortal orbit]

so

what's next

can you conjecture something for n=3?

[sharp wasp]

365(365-7)(365-7 - 7)/365^3

[mortal orbit]

there we go

[sharp wasp]

this is ugly though

[mortal orbit]

well it's what it is

(though fair warning : it's not the EXACT formula)

but it's a very very good approximation

[sharp wasp]

can u think something that will make it into a formula

[mortal orbit]

just a product

$\prod_{k=0}^{n-1}\frac{365-7k}{365}$

[potent lotus]

rafilou2003

[mortal orbit]

this formula gets worse and worse at approximating the probability as n gets bigger

but it should hold enough accuracy until we reach 50%

[sharp wasp]

it was simple

goddamn

u forgot 365^k

[mortal orbit]

I didn't

it's the denominator

365*365*365*...

[sharp wasp]

sorry

[mortal orbit]

anyways you get 48% with n = 9

[sharp wasp]

im exhausted

thats good

REALLY THANK U

<3333

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[gusty spade]

A country’s gross domestic production is modeled by the equation
P = 2L^3/5K^2/5 where L is the amount the country devotes to labor
and K is the amount spent on capital. Currently the country spends
5 trillion dollars on labor and 4 trillion on capital. If the country is
currently increasing their labor expenditures at a rate of 0.1 trillion dollars/year and their capital expenditures at 0.15 trillion dollars/year, at
what rate is the gross domestic production of the country increasing?

[carmine sorrel]

what have you tried?

[carmine sorrel]

I guess I otter change how I ask that.

[gusty spade]

tbh, I missed the online class where my professor taught us this type of problem so idrk where to start

& I'm meant to show each step and explain it

[carmine sorrel]

it's asking for a rate, do you know how to find the rate of something?

[gusty spade]

Is it the derivative?

[carmine sorrel]

yup. Find the derivative of P to find the rate

and since K and L are given as functions of time, you should use implicit derivation when you do

[gusty spade]

So I would take the derivative of the equation, then plug in values for K & L ?

[calm coral]

@gusty spade Has your question been resolved?

[gusty spade]

I got to here

Do I just plug in he values now?

[calm coral]

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[calm coral]

Please don't occupy multiple help channels.

[pure kayak]

you can try the rational root theorem

were you doing a division by (x-1)?

hm
| -x^3 -3x^2 0 4
x|-x^4 -3x^3 0 4x
-1 |x^3 3x^2 0 -4

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

just seems like a small error

-4 instead of +4

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[frosty mica]

is there a difference really between these two questions other than the greater than or equal vs greater than?

[calm coral]

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[frosty mica]

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[west monolith]

for part c of this question, do I just have to figure out $\frac{dz}{dt}=\frac{\partial{z}}{\partial{x}}\frac{dx}{dt}+\frac{\partial{z}}{\partial{y}}\frac{dy}{dt}$

[potent lotus]

Secret

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[idle fractal]

Why answer is C?

[calm coral]

@idle fractal Has your question been resolved?

[idle fractal]

<@&286206848099549185>

[blazing coyote]

why shouldn't it be continuous at zero?

[remote mural]

Does {.} Mean fractional part of x?

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[muted widget]

I don't understand where the "t" marked in purple goes

[pseudo wedge]

it's supposed to be in the same spot in the next line

that's a mistake

[muted widget]

okay

and where does the v_0 go?

[blazing hinge]

if its v_0t its added to the -1/2v_0t

[muted widget]

how

[blazing hinge]

you can see in the next line they got +1/2

so they added 1 to -1/2

[muted widget]

yeah I see that they did

but how

[blazing hinge]

wdym how

because they simplified

-1x/2 + 1x = 1x/2

[alpine stone]

There's a typo, should be v_0t instead of v_0

Ah it's been mentioned already

[muted widget]

in my head that would be 0/2

[muted widget]

thanks anyways though

[blazing hinge]

1x is like 2x/2

if you put in on the same denominator as -1x/2

[muted widget]

but

if we put x to 5

then
-((1 ⋅ 5) / 2) + (1 ⋅ 5) = -5/2 + 5 = -2.5 + 5 = 2.5

ah

I see

[blazing hinge]

yyyup

[muted widget]

thank you

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[runic wave]

how would you find the area of this ellipse

[runic wave]

otherwise how could i convert it to standard reduced form (because the position doesn't matter)

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[runic wave]

<@&286206848099549185>

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[runic wave]

<@&286206848099549185>

pls help someone

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[umbral wadi]

How to proceed?

[untold summit]

first write this in terms of 2^(-1/x) and 3^(-1/x), for example 4^(-1/x) = (2^(-1/x))^2

[umbral wadi]

After that...

[untold summit]

ok so you should have 9 (2^(-1/x))^2 + 5 * 2^(-1/x) * 3^(-1/x) < 4 (3^(-1/x))^2 which we can rearrange to get 9 (2^(-1/x))^2 + 5 * 2^(-1/x) * 3^(-1/x) - 4 (3^(-1/x))^2 < 0
now if we set a = 2^(-1/x) and b = 3^(-1/x) do you see how this is kind of like (a + b)^2 = a^2 + 2ab + b^2, just our coefficients are wrong

[calm coral]

@umbral wadi Has your question been resolved?

[umbral wadi]

Ok

I got it now Thank you so much 🙂

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[tall moon]

status 1

[idle marten]

Can you predict what shape the line would be

imagine y=x^1000 vs y=x

Non negative coefficients are important here

[tall moon]

no clue

like pointing upwards?

@idle marten

[idle marten]

Yes the function will be pointing up

imagine connecting 2 points , one is to the right and above the other

If you connected them with a y=x line, would that line be above a connection using y=x^2

Lowkey idk if I’m making sense lol

@tall moon

[tall moon]

i dont understand anythinf you said

not even a single word

[idle marten]

Do you know what a polynomial is

[winter elbow]

He definitely knows

[tall moon]

its like the f(x)=x^something+x^something+...... right?

[idle marten]

Yes so the highest degree of the polynomial will determine the shape of the line right

[tall moon]

like x^6+666x^4+2x^3/3-4x+3 right

[idle marten]

what are the shapes we can get

[tall moon]

dunno i never played with polynomials on a graph before

[tall moon]

i think like some sort of U shape and like a from down to up to down to up thing

like 2 candy canes stuck together on 0,0

[idle marten]

yeah i forgot how to describe those shapes but you can get, either a line, a u shape, or the cubic x^3 shape

quadratic shape

[tall moon]

yeah that goofy shape

[idle marten]

Ok so we know that the function will hit a certain y value at a certain x value

I think I’m missing some vocabulary

Imagine how each shape would approach that point

[tall moon]

dunno

[idle marten]

This is kinda what I came up with

So the line is at the top the U is the middle and the cubic is the bottom

what is the difference in the shape of x^2 and x^4

[tall moon]

like its more narrow?

[idle marten]

Yeah it’s more narrow

Hmmm I don’t really have a way to formally prove it but I can imagine the solution

[tall moon]

wait so to maximize f(12) a straight line is the best??

[idle marten]

yes

[tall moon]

o

[idle marten]

If you imagine x^1000

the line would be very steep

So the y value right before hitting the point would be a lot lower

Than the linear graph whose slope is less

[tall moon]

huh its still wrong

i got the graph 84x-480

at x=12 i get y=528

still wrong tho

wait a minute

its non negative coeficcients tho? @idle marten

a line has neg coefficients

[calm coral]

@tall moon Has your question been resolved?

[idle marten]

What part of the line function has a negative coefficient

what is a coefficient

[tall moon]

like say ax+b, a and b are coefficients

[idle marten]

Oh ur right

Ok well then fhe line isn’t possible butttt

[tall moon]

for a line, the graph would be 84x-480 which has negative coefficients

[idle marten]

that just means we need to increase the degree

you know I’m not sure what this topic is testing

I think you’ll end up with a quadratic where the line of symmetry is in the middle of the point tho

or a cubic function that’s weird

[tall moon]

cauchy-schwarz inequaligy

ok i checked and the answer used C-S

thanks for your time tho

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[daring kraken]

Is the part I circled not allowed

algebraically

[unique jackal]

it's allowed

there's nothing wrong with that

[daring kraken]

oh so the limit as n goes to infinity should diverge right?

[unique jackal]

correct

2^n outgrows ln(n)

[daring kraken]

i uhhh

am not sure what i'm doing wrong

unless the interval can be something other than 9

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[rocky cedar]

.reopen

n != -9 may be because of the ln(n)

[rocky cedar]

.close

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[remote mural]

tl;dr are "for any" and "for all" interchangeable??

[brittle isle]

yes

[remote mural]

cheers boss

why do we use two different ones?

[brittle isle]

just preference, it's our way of translating english into symbolic logic and back

[remote mural]

ah cool

thanks mate

.close

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[unique goblet]

find $\tan \frac a2$ if $28 \cot a + 27 \sin^{-1} a = 3, a \ \epsilon \ (3 \pi; 4 \pi)$

[potent lotus]

nashira._.

[unique goblet]

$28 \cot a + 27 \sin^{-1} a = 3 \ a \ \epsilon \ (3 \pi; 4 \pi)$

[potent lotus]

nashira._.

[pseudo wedge]

This question is very off

!original

[calm coral]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[unique goblet]

one sec

[unique goblet]

so we are just finding tan a/2

[pseudo wedge]

Huh

The question is very wrong

[unique goblet]

?

[pseudo wedge]

The domain of sin^-1 is [-1,1]

And (3pi,4pi) isn’t part of that

[remote mural]

Lol

[unique goblet]

this is not arcsin it is 1/sin a

[remote mural]

Bruh

[pseudo wedge]

That’s not how we usually write that..

[unique goblet]

well different places different conventions idk

coz in the book it is explicitly arcsin if they need it ig

[remote mural]

Alr let's say it is 1/sin

[unique goblet]

yep

so I tried to convert the cot into cos a/sin a and the sin^-1 to 1/sin a and got

$\frac{28 \cos a}{\sin a} + \frac {27}{\sin a} = 3$

[potent lotus]

nashira._.

[unique goblet]

then I collected them and got something like

$28 \cos a + 27 = 3 \sin a$

[potent lotus]

nashira._.

[remote mural]

Quadratic it would be

[unique goblet]

yep

so I squared both sides

[pseudo wedge]

You don’t need to do that

[remote mural]

Yeh then just replace cos and sin with tana/2

[unique goblet]

yeah using the tan half angle formula

[pseudo wedge]

You can write cosA and sinA in terms of tan(A/2)

[remote mural]

Yeah

To make it easy for eyes, use tana/2 as x
Look for the root which lies in the given interval

It's a/2 so keep that in mind

[unique goblet]

hmmm

im sorry im kinda dumb so i didnt get that at all

[unique goblet]

so how would we get tan(a/2)? Because yeah I get that we have the half angle formula. Would we find cos or sin or would we rearrange?

[pseudo wedge]

You don’t find either

[pseudo wedge]

Just write cosa and sina in terms of tan(a/2)

You can solve for tan(a/2) directly after that

[unique goblet]

$\tan \ \frac a2 = \sqrt{\frac{1 - cos a}{1+cos a}}$

[potent lotus]

nashira._.

[unique goblet]

so I rearrange for both cos and sin using the formula above?

[pseudo wedge]

No, not this

$$\cos a=\frac{1-\tan^2\frac{a}{2}}{1+\tan^2\frac{a}{2}}$$

[potent lotus]

kheerii

[pseudo wedge]

And $$\sin a=\frac{2\tan\frac{a}{2}}{1+\tan^2\frac{a}{2}}$$

[potent lotus]

kheerii

[unique goblet]

aaaah ok got it

[remote mural]

yeah

[unique goblet]

ok so I've solved for them and I have gotten

$\tan \frac a2 = 5 \ \ \tan \frac a2 = -11$

[potent lotus]

nashira._.

[unique goblet]

the second one is the correct answer so my final question is

since it is in the period of 3pi to 4pi wouldnt they be both correct since tan is both positive and negative there?

[pseudo wedge]

,w 28(1-t^2)/(1+t^2) + 27 =6t/(1+t^2)

[potent lotus]

Results provided by WolframAlpha

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

Requested by shikhargupta1207

[pseudo wedge]

No

tan a could be positive and negative, sure, but we’re talking about tan(a/2)

If a is in (3pi, 4pi)

Then a/2 is in (3pi/2, 2pi)

[unique goblet]

ooooh shit sorry im just a dumbass 💀

thanks so much for the help

[pseudo wedge]

No worries

[unique goblet]

.close

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[unreal flax]

When is
F( x ) = (x²+1)/(x²+3) Convex down

[remote mural]

you wanna find where is it convex down or what ?

[unreal flax]

Yes

[remote mural]

equate derivative to zero

[unreal flax]

Yeah i know

I get x = zero

[remote mural]

seems correct to me

(0,1/3)

(0,0.33333333)

[unreal flax]

I mean he wants a domain

Not a point

He didn't ask for local minimum

[remote mural]

graph of it

[unreal flax]

Yeah i know

So what's the domain when it's convex down

<@&286206848099549185>

Any help ???

[potent lotus]

Illiad

[unreal flax]

Yeah

[unreal flax]

It's when x < 0

F ' ( x ) = 0
When x = zero

Plug any number in the left of 0

F ' (x) < 0

Ok but the answer is wrong

[remote mural]

what is it then ?

[unreal flax]

It says it's convex down from [-1 , 1]

[remote mural]

i think the domain should be R

[unreal flax]

Me. Too

[potent lotus]

Illiad

[unreal flax]

So do you have any explanation for that?

Alright but the slope is negative for all x < 0

Then positive

[potent lotus]

Illiad

[unreal flax]

Why it's not relevant

Alright thanks

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[left vault]

ik im dumb but how do i find the gradient of these 9 secants it says i have to “use the equation to find the coordinate points and calculate the gradient” but idk what equation it’s talking about

[remote mural]

. In the convex quadrilateral ABCD given below, m (∠BCD) = 90°, |AB| = |AC| and AC ∩ BD = K. If the areas of triangles AKD and BCK are 10 cm² and 25 cm² respectively, how many cm² is the area of ​​quadrilateral ABCD?

A

10 cm

D

K

2

2

25 cm

B

C

A) 55

B) 60

C) 70

D) 105

E) 90

[left vault]

i think u have to find a new thingy channel cuz it’s occupied

[pure kayak]

the equation of f presumably

[left vault]

can you explain it in like dumb people terms because i’m only year 8 idk why i’m doing this work i’m confused

[pure kayak]

i assume you know the x values of the endpoints of the lines?

[left vault]

I know the start points but if i check with a ruler i can probably find the endpoint

[pure kayak]

can you show me exactly what was given to you?

[left vault]

Ok

Task 2

[pure kayak]

if youre approximating tangents the way you doing it is very unhelpful

tangents are tangential to the graph, yours are just horizontally across

when it says to the right, it means that if one of your x values was 4 for example, the end points might be about 4.3, 4.5, 5

[left vault]

oh

[pure kayak]

somewhere such that the line between them is at least in the same direction roughly as the graph

[left vault]

oh so like this

wait let me try it

i’m confused on how i would draw a line like that while still intercepting the graph in 2 places

would it be like the pencil lines

[pure kayak]

thats better, yes

you may want to change the scale of your graph though, its a bit compact

[left vault]

yeah true but for some reason the range has to be -20 to 30

[left vault]

it says here in task 1 idk why

actually nvm i can just make the domain from like -10 to 10 so it’s wider

thank you that heped

helped

[left vault]

one last thing, what’s the difference between task 2 and 3 like what would i do differently

it’s saying you gotta do a different method but it looks like the same thing just the other direction

[calm coral]

@left vault Has your question been resolved?

[calm coral]

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[deep island]

What is the indefinite integral of 1/x²-1dx

[royal sun]

Help

I dont get how to solve the question

[upper sparrow]

!occupied

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

[upper sparrow]

And of $\int \frac1{x^2} - 1 \dd x$, to make sure? Or did you mean $\int \frac1{x^2 - 1} \dd x$?

[potent lotus]

@upper sparrow

[calm coral]

@deep island Has your question been resolved?

[topaz raft]

cchhaart

[upper sparrow]

Waterrrrbeeeeaaaammmmmm

[topaz raft]

[calm coral]

@deep island Has your question been resolved?

[calm coral]

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[remote mural]

in prism ABC-A1B1C1, Plane AA1C1C ⊥ Plane ABC, AB=AC=BC=AA1=2, A1B=√6
a) Proof: AC⊥A1B
b) Find the sine value of the dihedral angle A-CB1-B

[remote mural]

<@&286206848099549185>

[hidden mantle]

!15min

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[remote mural]

sorry i am new here

[mint gust]

What is the question

[remote mural]

here

.close

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[vague grove]

how do i solve this

[vague grove]

because idk how to find part of a standard deviation

since mean + 1 standard deviation = 69 not 67

so idk how to find that

[calm coral]

@vague grove Has your question been resolved?

[wanton spear]

I'd just use the normal cdf function on your calculator

Or a giant normal cdf table
(Please don't, we made calculators do it for a reason)

@vague grove

[vague grove]

ok got it

where i can i learn how to use that

ill just look up a tut

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[cloud flume]

I guess this doesn't makes sense right

[near crag]

1-2(1-sin^2)
= -1 +2sin^2

[cloud flume]

So π/4-4x/3 i treat just like variable or

[near crag]

yeah

[vital raven]

$2\cos^2(x)-1=\cos(2x)$

[potent lotus]

Obotron

[cloud flume]

Got it

[cloud flume]

Got it

[vital raven]

so what you have is equal to: $-\cos \left( \frac{\pi}{2} -\frac{8x}{3}\right )$

[potent lotus]

Obotron

[near crag]

but you probably want to use a different trigonometric identity like Obotron said

[cloud flume]

Yeah

Just second

[cloud flume]

Now this seems...more appropriate I think

Bruh me and my camera

[vital raven]

yeh that looks correct

[calm coral]

@cloud flume Has your question been resolved?

[cloud flume]

Thabks

[calm coral]

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[meager spire]

to find the general solution

do i just take the determinant of this

with like lamda subtracted from the diag

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[swift laurel]

if you treat $x^3$ as its own variable (you can say for example $y = x^3$ if that's easier) then the polynomial is a quadratic and you can factor it as one

[potent lotus]

cloud

[swift laurel]

recognise by exponent rules that $x^6 = x^{3 \cdot 2} = (x^3)^2$

[potent lotus]

cloud

[swift laurel]

then we can make up some new variable, for example $u$, and say that $u = x^3$. then we have it that [ x^6 + 7x^3 - 8 = u^2 + 7u - 8 ]

[potent lotus]

cloud

[swift laurel]

that's the definition of a root, yes

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[uncut thicket]

If $(x_n)$ is a sequence with $\lim x_n = a$, how do I show that $x\left(\mathbb{N}\right) \cup {a}$ is closed?

[potent lotus]

tales

[uncut thicket]

Obviously every subsequence will have a as a limit

but it is not the case that any sequence in this set will be a subsequence of x_n

[gray temple]

True

So consider a sequence in the set which converges to something not in the set, notably distinct from a.

Can the original sequence still converge to a?

[calm coral]

@uncut thicket Has your question been resolved?

[uncut thicket]

yes, because you could take a sequence of decreasing indices for example

it it not necessarily a subsequence

[gray temple]

Well you can't decrease indefinitely

We're considering convergent sequences in the set

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[onyx quiver]

what do i do here knowing the normals are not scalar multiples?

hwo do i know if they interesect?

[swift laurel]

well either we have that:

1. the normals are parallel, so the planes are parallel, so they either don't intersect or are the same plane (their intersection is the entire plane)
2. the normals are not parallel, so they are not parallel and do intersect (their intersection is a line)

[onyx quiver]

so in this case its 2?

[swift laurel]

yes

[onyx quiver]

oh ok ty

.close

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[blazing moon]

How can I solve this from examples as I never done this before

[blazing moon]

I have nothing to draw on to show this as well

https://www.youtube.com/watch?v=0pqFytjlSDE

<@&286206848099549185>

[proper marlin]

what the

how is the point of intersection moving

[blazing moon]

Never done this one before

It is moving to the towards each toehr at 10 m/s

[proper marlin]

i get the question but idk what it means by "speed of the point of intersection"

theres two points of intersection

how is the point of intersection moving?

[blazing moon]

Just towards one another as tw points of intersection

Do you get it now

@proper marlin

I am getting a respone now on what is the meaning

The two circles are touching. Where the touch, that is the point of intersection. It would move up as the circles get closer.

[proper marlin]

Ok, so what I'd do is set the stationary circle to be $x^2+y^2=1$ and set the other circle to be $(x-2)^2+y^2=1$, so that way they are touching each other at exactly one point. Then, since the second circle is moving at 10m/s towards the other circle then the path of the circle is $$(x-2+10t)^2+y^2=1,$$ for $0\leq t\leq 0.2$ and so finding the point of intersection gives $$x=1+5t \quad \text{and} \quad y=\sqrt{10t+25t^2}.$$ Thus, the velocity in the $x$ and $y$ directions are $$\dot{x}=5 \quad \text{and} \quad \dot{y}=\frac{5+25t}{\sqrt{10t+25t^2}}$$ and so substitute $t=\frac{\sqrt{3}+2}{10}$ and find $|v|=\sqrt{(\dot{x}(t))^2+(\dot{y}(t))^2}$.

https://www.desmos.com/calculator/f9t4lu2xfq

[potent lotus]

one eight seven

[blazing moon]

So then just input the t then for the equation

[proper marlin]

at t=(sqrt(3)+2)/10, they are sqrt3 apart

yeah

[blazing moon]

0.37

[proper marlin]

idk probably

i was too lazy to calculate

[blazing moon]

7.31

sqrt(25+(5.34)^2)

Is the answer that I got

@proper marlin

[proper marlin]

idk is that correct?

[blazing moon]

Okay

<@&286206848099549185> I need to chekc calulcations for this here

at t=(sqrt(3)+2)/10, they are sqrt3 apart

I got .37 and then plug it in to get 7.32

sqrt(25+(5.34)^2)

@proper marlin they describe it as wrong

<@&286206848099549185>

@proper marlin it was 10 m/s as was solved with a different method

.close

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[blazing moon]

.reopen

[calm coral]

✅

[blazing moon]

.close

[calm coral]

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[harsh night]

Call a permutation $a_1, a_2, \ldots, a_n$ of the integers $1, 2, \ldots, n$ quasi-increasing if $a_k \leq a_{k+1} + 2$ for each $1 \leq k \leq n-1$. For example, 53421 and 14253 are quasi-increasing permutations of the integers $1, 2, 3, 4, 5$, but 45123 is not. Find the number of quasi-increasing permutations of the integers $1, 2, \ldots, 7$.

[potent lotus]

Aurora

[harsh night]

this sounds like a recursion problem

but im too stupid to think of the recursion

hints?

solved it nvm

.close

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[exotic umbra]

Someone can answer this?

[calm coral]

@exotic umbra Has your question been resolved?

[exotic umbra]

<@&286206848099549185>

[amber bolt]

assume there's 2 boys and 1 girl

then you can calculate the total masses of boys and girls, and add to get total mass of the class

[calm coral]

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[lime pike]

Hello. I have a question. Let's say there are five cards, and only one of them is winning card. You must pick a card randomly. If the card is winning card, you win and game is over. If you pick other card, you lose, put the card back, shuffle and give the deck to next player. There are three people playing this game, and each gets one chance to pick a card. What is the probability that the third player wins?

[lime pike]

Now I have two answers and not sure which one is correct.

One is 16/125 (4/5 * 4/5 *1/5) and the other one is 84/625 (4/5 * 21/25 * 1/5)

[brittle isle]

16/125 is correct

the third player can only win if the first two players lose

since each time a new player picks a card, the card is added back and the deck is shuffled, each player's draw is independent

[lime pike]

right. So I was confuses whether the probability of second person to lose is 4/5 or 21/25.

[brittle isle]

ah

actually

wait now i gotta think lmao

nah this is right

[lime pike]

ok

But I need an explanation

maybe I need to ask this question in discussion.

.close

[calm coral]

Channel closed

Closed by @lime pike

Use .reopen if this was a mistake.

[calm coral]

Available help channel!

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[hollow totem]

what have you tried?

[wet herald]

Expanding the bottom two

[hollow totem]

what does that mean?

[wet herald]

So xsquared + x

[hollow totem]

did you combine the fractions together?

[wet herald]

No?

How

[hollow totem]

the same way you subtract any other pair of fractions

[ancient thistle]

by writing them with a common denominator

[wet herald]

But how If x doesn’t have a common denominator

[hollow totem]

x doesn't even have a denominator at all

[wet herald]

And how would that show a+b if I work out the fraction

[ancient thistle]

what did you do here

[hollow totem]

it's the fractions that have denominators

[wet herald]

Expanded

[ancient thistle]

expanded what?

[wet herald]

The denominator

[ionic moon]

just cross multiply
$\frac{a}{b} + \frac{c}{d} = \frac{da + bc}{bd}$

[potent lotus]

esca (@ with reply)

[ancient thistle]

there's nothing to expand in the denominator here

i wouldn't call that cross multiplying

[wet herald]

Yes bro, x and x + 1

[ionic moon]

so what did you get

[wet herald]

I said before

Read higher up in the conversation

[ionic moon]

are you talking about this

[wet herald]

Yup

[ancient thistle]

that's not expanding

you're applying it incorrectly

that's how you find the common denominator

[wet herald]

But I didn’t now what to do

[ionic moon]

okay so

[ancient thistle]

write the fractions with a common denominator

[ionic moon]

you agree that
$\frac{1}{x} = \frac{x+1}{x(x+1)}$

[potent lotus]

esca (@ with reply)

[topaz raft]

bros got a million helpers

[ancient thistle]

,, \f 1 x - \f 1 {x + 1} = \f {???} {x(x + 1)} - \f {???} {x(x + 1)}

[ancient thistle]

you're supposed to do this

[wet herald]

That is literally halfway from expanding

[ionic moon]

just show us what you have then

[ancient thistle]

again, you're using the word expanding incorrectly

this is not "expanding"

this is rewriting the fractions with a common denominator, in preparation to combine them

[wet herald]

Ok I get it

Wait how did u make the other x(x+1)

What did u times both side by

[ancient thistle]

how do you make x+1 into x(x+1)

[wet herald]

Times by ohhh

Ok

I get it

But is that the answer

Bc it’s still not to square root

[ancient thistle]

no it's not the answer

you still have to solve the equation for x

this is just working towards making it easier to solve

[wet herald]

alr

So what now

[ancient thistle]

can you show what you have

[wet herald]

U want me to resend the question

[ancient thistle]

no i want you to show your working

[wet herald]

Ok

Hollon

I’m not great at working out on pc but here

[ancient thistle]

can you combine the fractions now

[wet herald]

Ye u get 1/x(x+1) = 4

[ancient thistle]

do you know how to solve this equation

[wet herald]

Nope

[ancient thistle]

do you know how to solve a simpler one like 1/x = 4

,, \f1x = 4

[wet herald]

Ye I think

[ancient thistle]

how would you do it

[wet herald]

Times by 1 to leave x

On its own

[ancient thistle]

times by 1?

[wet herald]

Yup

Oh wait

Nvm

If it was something other than 1 I could solve

[ancient thistle]

,, \f 2x = 4

[ancient thistle]

like this?

[wet herald]

Times by 2

[ancient thistle]

you sure?

[wet herald]

Bc it’s basically just 2 divided by x

So times it by 2

[ancient thistle]

,, \f2x \by 2 = 4 \by 2

[ancient thistle]

does this do what you want?

[wet herald]

Ye like that

Maybe

[ancient thistle]

so now it's [ \f4x = 8 ]

[wet herald]

No wouldn’t it be [ \fx = 8 ]

[potent lotus]

XVoid_76
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[wet herald]

No wouldn’t it be [ \x = 8 ]

[potent lotus]

XVoid_76
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[wet herald]

Bruh

U know what I mean

[ancient thistle]

,, \f2x \by 2 = \f4x

[ancient thistle]

this is how fraction multiplication works

like for example [ \f21 \by 2 = \f41 ]

[wet herald]

I swear if u times idk wich the denominator or numerator but it would get rid of the other

[ancient thistle]

think about it for a minute

1/2 times 2 is 1

[wet herald]

Bc they are dividing each other so opposite is to times them

To get rid of the other no?

[ancient thistle]

well yes

but you should remember which one is dividing which in a fraction

lest you need to go and review fractions

[wet herald]

No I get how to do fractions

[ancient thistle]

,, \f13 \by 3 = 1, \quad \f18 \by 8 = 1

[wet herald]

But I thought this was like the solv questions

Bro I know how to use fractions

[ancient thistle]

so what is [ \f2x \by 2 ]

[wet herald]

4/x

[ancient thistle]

okay so back to the equation

[hollow totem]

I promise you do not

[wet herald]

Hush

[ancient thistle]

,, \f 2x = 4

[ancient thistle]

how do you solve for x

[wet herald]

Idk

[ancient thistle]

you multiply by the denominator

[ancient thistle]

just as we do here

,, \f2x \by x = 4 \by x

[ancient thistle]

what does this become

[wet herald]

So u get 2 = 4x

[ancient thistle]

yes

so x is?

[wet herald]

Ohhh so the denominator is what is being divided

[ancient thistle]

i thought you knew how fractions worked

[wet herald]

Brother

[ancient thistle]

maybe you were lying

[wet herald]

Chill

I do

[ancient thistle]

you're the one making these bold claims

[wet herald]

If u have me a fraction question I could easily do it

I’m not in year 4

[ancient thistle]

i just gave you one

[wet herald]

No that don’t really count

[ancient thistle]

ok... anyway

[ancient thistle]

tell me what x is

[wet herald]

U need to divide by 4 tho bc x isn’t on its own

[ancient thistle]

yes so then x is?

[wet herald]

0.5

[ancient thistle]

damn not even a fraction

okay anyway that was just the warm up

back to the actual question

,, \f 1 {x(x + 1)} = 4

[wet herald]

F fractions

[ancient thistle]

how do you solve for x here

[wet herald]

Times by x (x + 1)

[ancient thistle]

looks like you've picked up the trick

what do you get

[wet herald]

Ye

x(x+1) = 4x(x+1)

[ancient thistle]

no!

[wet herald]

What lhs

[ancient thistle]

anyway faiyrose feel free to take over i gotta dip

[wet herald]

Bye

Hello fairy rose

Oh ok

I need to expand it huh

[potent lotus]

faiyrose

faiyrose

[wet herald]

Just c (x + 1)

I mean x

Not c

[potent lotus]

faiyrose

[wet herald]

1

1 = 1

[potent lotus]

faiyrose

[wet herald]

Oh that guy said differently

He said that he would be left with x(x+1) = 4x(x+1)

No

[hollow totem]

yeah that was you

[wet herald]

Hush boy

U is not here

[hollow totem]

you said that and then reacted to your own message with a checkmark

[wet herald]

Wizard relax

Ye continue @shell cliff

[potent lotus]

faiyrose

[wet herald]

So I expand the other side

Ok

4xsquered + 4x

[potent lotus]

faiyrose

[wet herald]

Is that it

Don’t I need a + b squarroot

How

[ancient thistle]

this is what's known in the trade as a quadratic equation

ever heard of it?

[wet herald]

Yes

I have done it before

I can’t subtract 1 on the rhs bc they all have x next to them

[potent lotus]

faiyrose

[wet herald]

Ok

Try and turn it into a + b squarerroot 2

Wich x needs to be on its own

I’m kinda lost at this point

Ye

I don’t remember it

But I know it

[potent lotus]

faiyrose

[wet herald]

Like that a + or - b squaroot a bunch of numbers

Ye what I just said

[potent lotus]

faiyrose

[wet herald]

Hmmmm I wonder

[potent lotus]

faiyrose

[wet herald]

Hmmmmmmmmmmm

Didn’t u flip the 0 to the other side

[hollow totem]

$ax^2+bx+c+0=$

[potent lotus]

thewizardofOU

[wet herald]

Tf?

That makes it worse

I get it it’s $4x^2+4x-1=0$

Ye my bad

[potent lotus]

XVoid_76

faiyrose

[wet herald]

The numbers above it

Brooo I’m not that stupid

Come on look

4 x squared is axsquared

Bx is 4x

C is -1

Yes bro

[potent lotus]

faiyrose

[wet herald]

Ok

[wet herald]

How I get this

[wet herald]

This

This

[potent lotus]

faiyrose

[wet herald]

It’s it left like that

[potent lotus]

faiyrose

faiyrose

[wet herald]

Not x (x+1) above and bellow

But how did u get that

Look this is my working out so far

But I don’t get how u then made the lhs a complete 1

I did

I made a common denominator

Oh wait

Hold on

I don’t change 4 yet until I need to get rid of the fraction huh

But still

I wouldn’t be left with 1

On the lhs

Bro I kinda gotta dip to school but thank u tho

Bye

.close

[calm coral]

Channel closed

Closed by @wet herald

Use .reopen if this was a mistake.

[calm coral]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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

[regal patio]

yo

[regal patio]

someone please help me learn whatever the fuck sin cos tan is

😭

[lilac nimbus]

sure

[regal patio]

okay thank you