help-49

[verbal solar]

which grade are you in?

[slender walrus]

same way you do long subtraction for something like
875 - 97

this just takes more effort

[verbal solar]

also, why in the world you would need to carry out such a huge calculation 😭

[slender walrus]

start small, look up a guide on long subtraction

[verbal solar]

learn that first then?

https://www.wikihow.com/Subtract

[last slate]

i was wondering if we could separate the number
example 1022 = 1000 +20+2

dk if that would make any sense

[verbal solar]

I mean both are the same things

[midnight plank]

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[cosmic stratus]

This might be quite long but I need help with my math project. Basically my idea is to take a screenshot from google maps of a motorway interchange, scale it on GeoGebra and then fit a curve to the road. We can then find the centripetal force and find the maximum velocity of the car before it slides off. Since the road is designed as part of a Euler spiral I then wanted to use fresnel integrals to show this and the constant acceleration. Would this work and how complex is the math for fresnel integrals? Also if we fit a curve but in reality it is an arc how major is the difference? I’m basically asking whether this project could work?

[cosmic stratus]

It’s a maths project and I need to show complex math. Would this work mathematically?

[carmine sigil]

@cosmic stratus did you manage to see what I wrote in your last thread?

#help-32 message

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[chilly sky]

help

[chilly sky]

I am working on boolean algebra and need some help with simplification

this is the expresssion

I am told to "Use any Boolean properties/theorems you know to simplify it"

[midnight plank]

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[chilly sky]

@stiff heart

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[iron escarp]

Anna bought 2 books, the probability that she likes the first is 0.5, and prob that she likes the other is 0.5. The probability that she likes both is 0.3. What's the prob that she likes neither of the books? So let A=likes the 1st, B=likes the 2nd, $A \cap B=$likes both, then we are asked to find $ P(A^C \cap B^C) =P((A \cup B)^C) = 1 - P(A \cup B) = 1 - 0.7 = 0.3 $. Is this correct?

[grand pond]

atif

[misty carbon]

ye

[iron escarp]

thx, and let me ask

what if i have two independen events

how do i find P(A \cap B) is it just P A * P B ?

[misty carbon]

$P(A \cap B)$

[grand pond]

Heartuary

[misty carbon]

ye

[iron escarp]

but if they are dependent then we can't find out right? given only the probs of A and B

[grim vector]

No its union ? Or intersect ?

[iron escarp]

dependend**

[misty carbon]

ye

[iron escarp]

wdym $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

[grand pond]

atif

[grim vector]

When independent, intersection is empty no ?

[misty carbon]

no

[grim vector]

Ok mb

[iron escarp]

ohhh

@grim vector u confused it w disjunct events

thank you i realised now the distinction

[grim vector]

Yeah, cuz i call it independant

In my langag

[iron escarp]

was this a confirmation for the question about the dependent events case, that we dont know the prob of \cap

[misty carbon]

ye

[iron escarp]

thank you very much

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[gentle pebble]

Can someone explain me what dz -> dt * dt/dz stands for

[gentle pebble]

<@&286206848099549185>

[pearl idol]

Please don't ping Helpers unless your question has been unanswered for 15 minutes

[gentle pebble]

Ok

[night egret]

they simply parameterized it

[gentle pebble]

Could you elaborate further

What does it mean

[night egret]

instead of z being a variable, they made it into a function dependent on another variable

[gentle pebble]

so it is f(z(t) ?

[night egret]

yes, as you see in the second integral in example 9.6

[gentle pebble]

But why at 9.1.1 it’s declared twice times

dt*dz/dt * f(z(t)

[night egret]

what specifically is declared 2 times?

the dt?

[gentle pebble]

Yes

[night egret]

it's cuz of chain rule

cuz t has it's own independant rate of change

[midnight plank]

@gentle pebble Has your question been resolved?

[sage olive]

you might want to reference calc iii where line integrals were parametrized to turn into definite integrals wrt the parameter

think of it as a big u substitution, you have the integral of f(z(t)) * z'(t) dt. if you let u = z(t) then du = z'(t) dt, so your integral is f(u) du

it's just the other way around, letting z be a function of t now requires accounting for the rate of change wrt t

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[gusty sinew]

I need help with my math test

[sage olive]

it's against the #rules of the server to ask for help on graded assessments

[flat hollow]

ayo ⏰ ⏰ ⏰ ⏰

[gusty sinew]

its not graded

its s/n

[coral harness]

you use inverse trigonometric functions here

we know the tangent of theta in (a) is 19/24, so find the arctan of 19/24

the sin of theta in (b) is 97/131, so find the arcsin of 97/131

[midnight plank]

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

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[viral dagger]

let S be a set {0,1,2,...,9}, a subset of S is formed that has atleast 3 elements, what is the probability that the subset in that atleast has 2 numbers that are a multiple of 3 and the sum of all the elements are a multiple of 3

[viral dagger]

i have an answer and i wanna get it checked

so the ammount of ways to form the subset is 2^10 (total subsets) -1 (empty subset) -10(single element subset)-2c10 (two element subset)=969

for the ammount of subsets that fit in the requirements i wrote S in mod 3, so {0,1,2,0,1,2,0,1,2,0}, for the first requirement, it would be 2^4 (total zeroes)-1 (not picking any zeroes) -4 (picking only one)=11

for the second requirement it can be
none
111
222
a combination of picking 1 and 2, picking two of 1 and 2, picking three of 1 and 2

so
none (1 possibility)
111 (1 possibility)
222 (1 possibility)
12 (3x3=9 possibility)
1212 (2c3×2c3=3x3=9 possibility)
121212 (1 possibility)

so 22

both of these are independent of eachother, so its fine to multiply them

11×22=242

however, theres a chance of picking only two 0's and picking none of the 1 and 2's, so subtract 4 from 242 to get 238/969

is this correct?

oh wow you can actually simplify them

14/39

any flaws in this?

[visual tiger]

Wait 2^10 - 11 = 969?

[viral dagger]

wait

[visual tiger]

You forgot to subtract 2 elements

[viral dagger]

yeah

[visual tiger]

So -45

[viral dagger]

yeah

noo fuckk the denom is wrong :((

[visual tiger]

1024-56 = 968

[viral dagger]

damn i didnt get a single question right bruh

other than that?

[visual tiger]

Uh so

For the 0,3,6,9 elements you have indeed 11 choices

And then for the 1,2,4,5,7,8

You either pick none (1), all (1)

1 of each (3×3)

Or 2 of each (3×3)

Or 1,4,7

Or 2,5,8

So 11×22 = 242

242/968

[viral dagger]

wb when picking 2 elements from 0369 and picking none

[visual tiger]

.

Oh right

So - 6

236/968

[viral dagger]

that damned one >:(

thanks again

.close

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[viral dagger]

down to one question woohoo

[visual tiger]

59/242

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[rotund jacinth]

I don't understand how you're able to do this? Is this explainable by a venn diagram? The age kinda throws me off. I'm not really sure how to tackle this problem...

[rotund jacinth]

I know that since the events are independent you can do P(A)*P(B) = P(A and B) but I'm not sure why you're able to yoink 21/140 and 80/140 for these values. Why is P(A and B) = a/140 in other words

[misty carbon]

thats just the definition of a contingency table

[midnight plank]

@rotund jacinth Has your question been resolved?

[idle veldt]

Think of it as asking what is the probability of someone in the 140 ppl liking tennis and 35 and younger, this is a/140 where a is the amount of ppl who fit under this category
So to find the probability its the probability of the person liking tennis and the person being 35 and under

The probability for these two events are just the pop. of the group over the whole pop.

So 21/140 comes from 21 ppl who like tennis over the 140 ppl surveyed

[rotund jacinth]

Ah I kinda see

[rotund jacinth]

but what about 80/140?

[idle veldt]

The chance for someone whos 35 and younger in the total of 140 ppl

[midnight plank]

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[idle veldt]

Shit i mean

[rotund jacinth]

.reopen

[midnight plank]

✅

[rotund jacinth]

Hmm so if 21/140 is the people who like tennis and are 35 years or younger and 80/140 is merely the probability of someone who is 35 years or younger, why are they treated as the independent events which get multiplied?

I'm kinda confused at this part

AHH, I see, thank you nevermind, "its the probability of the person liking tennis and the person being 35 and under"

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[runic sage]

Calculate the distance from the point (10, 1, 1) to II.

[runic sage]

i dont understand how to solve this

i have the II which is 2x-y+z=3

but i dont understand how to calculate the distance from the point

i know that we need a another point such as that we have been given

but how to get a another point i have no clue and need help with that part

in the answer they have got the point (0,0,3)

which i rly dont understand how

<@&286206848099549185>

[modern sapphire]

how would you go about finding the distance

more specifically what is the criteria for the distance to be "shortest"

[runic sage]

thats the part idk

[modern sapphire]

so your line is a collection of points
and there is only one that qualifies to be closes right?

so maybe draw a diagram for a random line and a random point and show which is closest one

[runic sage]

yes but what i understand is that there is also other point which can be used for (10,1,1) right?

beside (0,0,3)

[modern sapphire]

Hint: || the closest point has the perpendicular distance as the minimum ||

[runic sage]

i dont understand how this helps lol i am still clueless

or rather i am lost

[modern sapphire]

so you have to find a line that is perpendicular to the given line

[runic sage]

yes

[modern sapphire]

and also passe through the given point

[runic sage]

yes

[modern sapphire]

so can you find that line?

[runic sage]

no

[modern sapphire]

you have the slope and the point

there is a standard formula for that

[runic sage]

wait do we have the slope?

[modern sapphire]

you have the whole equation of the main line

[modern sapphire]

right here

[runic sage]

uhhh okay

so the slope in this case is 3?

[modern sapphire]

no

technically there is infinite perpendicular line, so you have to take a general line and do the dot product

and also pass it thru the point

and solve for the coefficients

since it is in 3d

[runic sage]

can we go step by step how to solve it ?

[midnight plank]

@runic sage Has your question been resolved?

[midnight plank]

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[viral dagger]

hey, just curious, how would you find the length of a line of a parabola, like say y=x^2 and the length of the line from (0,0) to (4,16)

[viral dagger]

im guessing this uses calculus but i only know the rudementary things

[dreamy glade]

what does this mean? like the gradient/ slope?

[viral dagger]

uh

[dreamy glade]

or r u cutting a line through the parabola to those points

[viral dagger]

like the length of this line

[dreamy glade]

oh

[viral dagger]

i saw some question about cables and made me wonder about finding this

[dreamy glade]

that makes a lot more sense, srry bout that

https://www.youtube.com/watch?v=3P3oxDc6hUs this video might help, tho it is a little advanced

u might wanna first catch up on some trig and calc before attempting this, but if u are fine with that, then this vid should be good

[viral dagger]

wtf 😭

i only know a hint of trig lmao

[dreamy glade]

some other helper might come in and explain it a lot easier, so ig the best option is to wait and learn some trig and calc while waiting

[viral dagger]

yeah this is wau to complicated

[dreamy glade]

[silk hound]

What about Pythagorean theorem?

[viral dagger]

first minute have no idea whats going on 🙏

[dreamy glade]

that isn't trig

[viral dagger]

what

[silk hound]

So real

[viral dagger]

alr thanks

.close

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[silk hound]

But intuition for arc length integral can be provided through Pythagorean no?

[viral dagger]

.reopen

[midnight plank]

✅

[viral dagger]

go on

[silk hound]

One sec

subdivide the interval

the green lengths are an approximation of the length of the red curve

@viral dagger

then sum over all of the intervals

[viral dagger]

uhhh

how can i find dy/dx?

[silk hound]

if for example y = x^2

then dy/dx is just 2x

[viral dagger]

is it just the derivative?

[silk hound]

yea

[viral dagger]

huh

i thought derivative is d/dx? or did they just cut the y out

[silk hound]

i like to view this as verb vs. noun

d/dx means to take the derivative of whatever appears to the right of itself with respect to x

dy/dx is the derivative of y with respect to x

[viral dagger]

erm ok

[silk hound]

but yeah

from here, as the intervals become more dense

the change in x approaches 0

so delta y / delta x approaches dy/dx

and the summation approaches an integral

hopefully that should make this formula look a little bit more intuitive

[viral dagger]

so for this, would it be $$\int^{4}_{0}\sqrt{1+2x} dx$$

[grand pond]

Skill_Issue

[silk hound]

keep in mind that u have to square the derivative

[viral dagger]

oop

[silk hound]

$$\Delta s=\int^{4}_{0}\sqrt{1+(2x)^2} dx$$

[grand pond]

Really?

[viral dagger]

err so $\int^{4}_{0}\sqrt{4x^2+1} dx$

[grand pond]

Skill_Issue

[silk hound]

yes

[viral dagger]

funny thing i have no clue how to integrate square roots

[delicate helm]

i actually don't remember either is this a trig sub of some kind?

[viral dagger]

soo

[delicate helm]

or is this elliptic?

[silk hound]

Well in this case u could use a trig sub

[viral dagger]

the $\int \sqrt{4x^2+1} dx$ is $$\frac{(4x^2+1)^{\frac{3}{2}}}{6}+c$$?

[grand pond]

Skill_Issue

[delicate helm]

no, right?

[viral dagger]

i have no idea what a trig sub is

[delicate helm]

cuz chen lu

[silk hound]

Not possible

[viral dagger]

huh why :(

[silk hound]

Ye

[violet storm]

taking the derivative of your answer will convince you that doesn't work because you don't just have x^(1/2) you have f(x)^(1/2)

[viral dagger]

what

[delicate helm]

x = tan(θ)/4

[viral dagger]

wtf

[violet storm]

basically you need a u sub, but you don't have the stuff necessary to build the du as in some cases

[viral dagger]

erm

is it bad i dont know what a u sub is

[silk hound]

Well that depends on where u are in math, no?

[violet storm]

well that's one of the first things you learn as far as integration, so are you just starting to learn integration?

you shouldn't be learning trig subs yet

[viral dagger]

i watched a video on it, i was kinda hoping the integration wouldnt be too complex

[violet storm]

it is, it's a lot more difficult than differentiation. differentiation is basically like once you know the algorithm you can differentiate anything, but integration actually requires being familiar with certain patterns, techniques, and there will be many expression that just aren't integrable

[viral dagger]

oh

[silk hound]

😭

[viral dagger]

i was kinda banking on this, why doesent this work tho?

[silk hound]

the inner function being a linear function

makes the chain rule aspect pretty easy

we don't have that

[viral dagger]

oh

[violet storm]

because you don't have an ax+b expression your x has an exponent

this sums up differentiation vs integration pretty well:

[viral dagger]

ok fair enough

thanks

[silk hound]

should i unironically try to miserably integrate this?

[viral dagger]

.close this is way out of my league, ty everyone

[midnight plank]

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[last slate]

is the question faulty?

also well sample > population right?

[night egret]

population is every individual

sample is the selected individuals from the population

sample <= population always

[last slate]

isn’t sample divided by n-1?

[night egret]

the sample standard deviation is

but there’s no guarantee that it’s smaller

[modern sapphire]

they asked if pop sd > sample sd

[last slate]

isn’t sample sd >= population sd?

[modern sapphire]

you seem to be saying sample > pop which makes your marked answer opposite of what you say

[last slate]

that’s not my answer (it’s my friends)

my teacher said the question is faulty

[modern sapphire]

oh lol

[last slate]

but why?

[modern sapphire]

yea it is faulty, coz sample sd is no a constant

it depends on sample which you can choose to be either ways

[last slate]

but x/ (n-1) > x/n for positive x right?

[modern sapphire]

but n of populatoin is not same as n for sample

also one uses sample mean and other uses pop mean which are also different

[last slate]

oh lol

[modern sapphire]

Also thats why there exists things like hypothesis tests

[last slate]

okay so we can’t tell right?

sample sd could be >= pop sd or < pop sd right

[midnight plank]

@last slate Has your question been resolved?

[midnight plank]

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[dreamy lichen]

What's the minimum n, such that in a n x n board, divided into black and white unit squares, we can always find a rectangle (consisting of those unit squares) whose corner squares are of the same color

[dreamy lichen]

I'd appreciate few hints

[last slate]

are the colored squares arranged like that in a chess board ?

[junior flower]

w b w
b w b
w b w
appearing is already enough, so it’s at most 3

[dreamy lichen]

no, any coloring

[junior flower]

oh

[dreamy lichen]

in arbitrary coloring, sorry I forgot to mention that

[junior flower]

this question doesn’t make much sense to me then…

oh nvm it does

[last slate]

sounds like a pigeonhole problem

[dreamy lichen]

yeah, it kinda does

except i dont see the pigeons and holes

[subtle zinc]

you can solve this by brute force

[dreamy lichen]

I'd still need a proof

[subtle zinc]

that is proof

[dreamy lichen]

ideally one that doesnt include computers

I'd have to consider all the colorings

possibly exclude few symmetrical ones

and that's too complicated to do by hand

[subtle zinc]

oh god, symmetry

there is way too much symmetry

[last slate]

n is in units, right ?
and the small boxes in it are of a unit

[dreamy lichen]

yes

and the rectangles must have sides parallel to sides of the board

and must consist only of the unit squares

[last slate]

if we consider a single block and give it any color, would that be fine ? 🤓

[subtle zinc]

w b b
b w b
w w w

i think this has no such rectangle

[dreamy lichen]

Oh, I forgot to mention this. The sides of the rectangle must be at least of length 2

[dreamy lichen]

yes, im pretty sure i even made 4 x 4 one with that property

[subtle zinc]

ok good

[dreamy lichen]

oh wait, I didn't

it has a rectangle I didnt notice at first

maybe i can fix that

im pretty sure it's fixed now

[junior flower]

mayervietorus

[dreamy lichen]

w w b w
w b w b
b b b w
b w w b

hmm, if i think about it arrangement of columns and rows doesnt even matter

[subtle zinc]

sup?

[junior flower]

nothing just saying hi

[subtle zinc]

hi

[junior flower]

Error_5506

[dreamy lichen]

is typing...

[dreamy lichen]

I think this can get me an upper bound on n at least

[carmine void]

Yes, I believe we can say wlog that we can move columns and rows it like this

w w w b
w b b w
b b w b
b w b w

[subtle zinc]

do you know such an n exists btw

[dreamy lichen]

well

I dont

but im fairly sure it does

[visual tiger]

Oh 5 always works

[dreamy lichen]

You got it?

[visual tiger]

Think about the possible 2 first colors for each column

[dreamy lichen]

Just a hint please

[dreamy lichen]

okay, thanks

Ohh right

[shell wigeon]

Did I make a mistake?

[dreamy lichen]

There are 4 possibilities, bw, wb, bb, ww

[dreamy lichen]

hmm

[subtle zinc]

bbw

[junior flower]

that has a white rectangle

[shell wigeon]

Oh, so it does

[dreamy lichen]

by pigeonhole, one of them must be there twice

[carmine void]

Can we make the top left black?

[junior flower]

no

then there is a black rectangle

on the far left side

[dreamy lichen]

If it's ww or bb, then we are done

[visual tiger]

It's a good start, but afterwards we need 4 to have the same color

[dreamy lichen]

otherwise it's bw or wb

hmm

[last slate]

🤔 idk

[dreamy lichen]

I think I got it

[last slate]

sorry guys, i dont really know what pigeonhole is and maybe i am not capable enough.
will just watch

[dreamy lichen]

If there are 5 squares of same color in a column, then there is 10 pairs of same color in that column. If there are 4, then there is 6. If there are 3, there is 5.
By PHP, one of these always happens. So there are at least 5 pairs of same color in every column.

and there is total of 10 pairs in each column

wait this wont work probably

[visual tiger]

I think I have another idea

[dreamy lichen]

tell me

[visual tiger]

Take the first column

It always has 3+ squares of the same color

Say it's black

[shell wigeon]

Oh yeah that's a good one

[dreamy lichen]

are you gonna construct a rectangle 2 of whose corners are in first column?

[visual tiger]

Bot necessarily

[dreamy lichen]

oh, okay

[visual tiger]

But only look at the 3 rows with those 3 black squares (so 3*5 rectangle)

If there's another column with 2+ black squares it's over

[dreamy lichen]

why should all of them be black

[visual tiger]

If they're white just reverse the colors

[dreamy lichen]

this will have to reverse too then

which is not always possible i think)

[visual tiger]

No I'm saying

Look at the first column

Say

w
B
B
W
W

No matter the 5 colors

[dreamy lichen]

oh wait i see what you mean

[visual tiger]

There's always 3+ of the same color

If it's white just switch all colors in the square

So
B
W
W
B
B

[dreamy lichen]

right

[visual tiger]

Now we get rid of the rows with white

[dreamy lichen]

2+ means >2 or >=2?

[visual tiger]

=2

[visual tiger]

So now we restrict ourselves to the 1st, 4th and 5th row in this example

B - - - -
B - - - -
B - - - -

[dreamy lichen]

what if there was one with 2 black squares
B W
W B
W B
B W
B W

[visual tiger]

No we restrict ourselves to those rows

[dreamy lichen]

oh

i see

[visual tiger]

What happens to the rows without black in first we get rid of

[visual tiger]

So if in this rectangle, there's another column with 2+ black squares...

[dreamy lichen]

then its over

right

[visual tiger]

Yes

[dreamy lichen]

si there can be at most 1 black square

[visual tiger]

Per column

So there are always 2 (or 3) white squares in each of the 4 remaining columns...

[shell wigeon]

visual aid, get rid of the red rows (don't care what color they contain), you can't place 4 blacks in the remaining 3 rows such that there is no white rectangle (in blue here)

[dreamy lichen]

wait yes, there can be 3

white

[modern sapphire]

It seems to me we can generalize this to n^2+1 sided squares fr n colors of tiles

or am i wrong

[visual tiger]

At least 2

So what are the possible columns

[dreamy lichen]

W B W W
W W B W
W W W B

there is 4

[visual tiger]

4 possible columns

[dreamy lichen]

4 pigeonholes

but if there is the WWW one

[visual tiger]

If www we win

[dreamy lichen]

then it will form rectangle with one of those other

[visual tiger]

If not

[dreamy lichen]

so 3 possible columns

[visual tiger]

4 columns

[dreamy lichen]

4 pigeons

right

and so we win

[visual tiger]

👍

[dreamy lichen]

that's a nice approach

thanks

how do I learn to come up with these?

[shell wigeon]

What's a n^2+1 sided square?

[dreamy lichen]

i knew I have to use PHP, but i just didnt know how

[dreamy lichen]

that's quite interesting generalization

[modern sapphire]

n^2 + 1 by n^2 + 1 blocks that form the square board

[dreamy lichen]

then there would be at least n+1 tiles with same color in first column

[subtle zinc]

wait so what's the answer

[dreamy lichen]

5

[subtle zinc]

cool

[dreamy lichen]

hmm

i wonder if i could make stronger claim

and claim there are at least 2 rectangles

or more

anyway, ill probably think about that myself

thanks for all the help

.close

[midnight plank]

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[shell wigeon]

I don't think that's true due to symmetry

[dreamy lichen]

.reopen

[midnight plank]

✅

[dreamy lichen]

why?

why due to symmetry?

my thought was that I could probably repeat the argument for a different column

and get a different rectangle

[modern sapphire]

where is the 2nd?

[dreamy lichen]

flip that one

[zealous schooner]

this sounds like an interesting question

[dreamy lichen]

and you get the other one

(around y = -x)

[modern sapphire]

flip? you mean turn white to black and vice versa?

ohh

[zealous schooner]

i mean, does that really count as different rectangles?

[dreamy lichen]

i meant this

[zealous schooner]

ahhh

right

[modern sapphire]

oh yea

what if that rectangle were square

[shell wigeon]

That's what I'm trying to make

[dreamy lichen]

that's also interesting

damn this question has quite a lot to offer

[zealous schooner]

I didn't really understand the solution

mind explaining it to me a bit?

[dreamy lichen]

sure

So take the first column

there must be 3 tiles with same color in that column

wlog, say it's black

now we restrict ourselves to only the rows with those 3 black tiles in 1st column

visual from Nel

So we now have 3 x 5 board basically

[zealous schooner]

ahh I was confusing rows and columns

[dreamy lichen]

in the other 4 columns, there cant be more than 1 black tile per column

lest it would create a black rectangle with the first column

[zealous schooner]

right okay

[dreamy lichen]

and so there is either 0 or 1 black tile

this leaves us with:

W B W W
W W B W
W W W B

[zealous schooner]

wait what are the blue tiles here

[dreamy lichen]

i think its just marked rectangle

white

[zealous schooner]

ahhhh

I see

any set of two configurations from this will have atleast two matching rows

that's neat

[dreamy lichen]

so those are the 4 possibilities for our remaining 4 columns

[dreamy lichen]

really?

B W
W B
W W

[zealous schooner]

oh, no

nevermind

[dreamy lichen]

but what we can say is that none of them can be repeated

so each column gets exactly one of those

but the white column will make a rectangle anyway

so there must be a rectangle

[zealous schooner]

ohh lmao

yeah

nice

basically just PHP

[dreamy lichen]

very nice, i wouldnt have thought of this solution myself

[zealous schooner]

in a smart way

[dreamy lichen]

yes

and i struggle with that

I knew it's PHP, but i cant apply it smartly enough

[zealous schooner]

I think there might be an easier way

actually nvm this seems simple enough

[dreamy lichen]

I wonder if i could cook anything with this

[zealous schooner]

yeah I was actually thinking something pretty similar to this

but it just doesn't seem to work out..

unfortunately

5 of same color - 10 pairs
4 of same color - 4c2 = 6 pairs
3 of same color - 4 pairs no?

[dreamy lichen]

Actually i probably could

but it would include some case work

nvm i cant

thanks again everyone!

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[dreamy lichen]

I completed it. There will be at least a total of 25 pairs of tiles in same column with same color. But there are in total 10 possible pairs of tiles in one column. So by PHP, there are at least 3 columns, such that they all have a pair of same type, that's colored in a single color per column. But at least 2 of the 3 columns must use same color for coloring of their pairs. And those form our rectangle

far more complicated, but works

and i also struggle to word it

[shell wigeon]

I think I found a much simpler proof, and also one that shows you always have two rectangles

[dreamy lichen]

can you show it?

[shell wigeon]

Among the 5 columns, at least 3 will have 3 squares of the same color

[dreamy lichen]

wait what

oh

[shell wigeon]

[midnight plank]

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[foggy tartan]

why is this wrong?

[foggy tartan]

I want to proof for volume of sphere

[midnight plank]

@foggy tartan Has your question been resolved?

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@foggy tartan Has your question been resolved?

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[wide marsh]

how do you find the point on a parametric function with a specific slope

ive already found dy/dx but dont really know how to find a point from that

this is the profs answer, can anyone explain why

are the points literally just (x, y) where x=4cost and y=4sint except for t you write arctan(-2) as i found earlier

[rose trout]

You've found the "time" at which the slope is 0.5, and the parametric equation of the curve gives you a point for every instant.
So yes, you can just plug the value you got for t in the parametric equations, and this will give you the point at which the slope is 0.5.

[midnight plank]

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[last slate]

dababy

[last slate]

help

dancing in the moonlight

aloo

[last slate]

yoooo hayley

okay

so first

it says no solution

so equation 1 must equal equation 2 right

soo what i did is

i made y the subject to both equations

[olive matrix]

not necessarily

[last slate]

what

[olive matrix]

in fact, if the two equations equal each other, the system will have many solutions

[last slate]

oh

[olive matrix]

eg
x + y = 1
x + y = 1

[last slate]

ohh

so what should i do?

[olive matrix]

well, let's look at it from both sides. first, can you think of a system of equations that has no solution?

[last slate]

OHH

parallel?

so

[olive matrix]

yep

[last slate]

wait lemme try

go drink water and come back

[olive matrix]

okay :)

[last slate]

p=6

so basically

i made both y subject

the slope on eq 1

1/12

slope on equation 2 0.5/p

so 1/12=0.5/p

so p =6

damn thats so smart

[olive matrix]

yep, you could do it a few ways but solving for y works just ifne

[last slate]

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[nova yoke]

cross product?

[tidal turret]

between the direction vectors

,, \left(1,1,0\right) \times \left(0,-1,-2\right)

[grand pond]

938c2cc0dcc05f2b68c4287040cfcf71

[midnight plank]

@tidal turret Has your question been resolved?

[tidal turret]

<@&286206848099549185>

can someone help my fatass

[nova yoke]

close but not exactly right

should be (-2, 2, -1)

don't forget that the j cofactor gets a minus sign

[tidal turret]

why cross product

[nova yoke]

wdym

[tidal turret]

it creates a perpend9icular vector

to both direction vectors

[nova yoke]

yea, the cross product of two vectors is perpendicular to both vectors

it's a simple way to find such a vector

[tidal turret]

now what

[nova yoke]

so now you have the direction of the line L3

you just need a point on L3

[tidal turret]

how do I find the point

[nova yoke]

you know you want L3 to intersect L1 and L2

so find where L1 and L2 intersect

that's the point you want

[tidal turret]

,, \lambda(1,1,0) + (0,0,2) = \lambda(0,-1,-2) + (1,-2,-4)

[grand pond]

938c2cc0dcc05f2b68c4287040cfcf71

[nova yoke]

except be careful

it won't necessarily happen at the same lambda for both lines

so you should do something like:

$$\lambda(1,1,0) + (0,0,2) = \mu(0,-1,-2) + (1,-2,-4)$$

[grand pond]

Bungo

[tidal turret]

fair

mmm

[nova yoke]

yea so that's three scalar equations with two unknowns

[tidal turret]

is a system of equations

[nova yoke]

yep

[tidal turret]

but

I dont get it

howt ot turn it into system

[nova yoke]

just do it component by component:

alpha(1) = gamma(0) + 1

alpha(1) = gamma(-1) - 2

alpha(0) = gamma(-2) - 6

the third one easiest since the coefficient of alpha is 0

[nova yoke]

ok

so alpha is 1 and gamma is -3

and the third equation gives 0=0 which is consistent

(if it was inconsistent that would mean that the lines don't intersect)

[tidal turret]

fair

[nova yoke]

so now you can find the point of intersection

[tidal turret]

how

I need more handholding

[nova yoke]

you can either plug alpha = 1 into the equation for L1, or plug gamma = -3 into the equation for L2

you'll get the same answer either way

[tidal turret]

gotcha

[nova yoke]

into these, i meant:

[tidal turret]

I feel u

[nova yoke]

ok

so you have the point of intersection, and you want that point to be on L3 as well

and you also have the direction of L3

[tidal turret]

we got L3 fully

[nova yoke]

yep

[nova yoke]

looks good assuming that middle stuff is your answer

[tidal turret]

purple

.solved

[midnight plank]

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[rich plaza]

How do I do this

[rich plaza]

I mostly understand how vectors work

but what do I do when I have the d=3c=(9,12)

also the video they gave me, it doesnt work

[sharp coral]

they give you that d = (9,12) so you can treat it like any other vector

[rich plaza]

oh ok

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[red cave]

how do i solve this

[atomic magnet]

there may be a better way, note that alpha, beta and tan^-1(whatever) are all angles, so I would probably let the expression above be equal to some theta, and take the tangent of both sides. so, tan([a+b]+tan^-1(....)) = tan(theta). then, simplify the LHS using tangent sum formula, and just keep simplifying to get the expression. also later it would be useful to draw a triangle using SOHCAHTOA

i think this is the best way, but there may be smn quicker

[red cave]

what do you mean take the tangent of both sides?

isnt there only one expression

which parts

yeah

[atomic magnet]

say we are trying to simplify an expression a + b + c, which we set equal to some X, i.e. we set a+b+c=X. Then, we can simplify by first considering tan(a+b+c)=tan(X), and then taking the inverse tangent at the end

we will get some cancellations along the way

[red cave]

Idk what the question even is

[midnight plank]

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[visual comet]

can someone check this pls

[feral sedge]

i agree with that answer

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[visual comet]

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[dreamy lichen]

So I tried solving it this way:
What I need to calculate is equal to
|M3 - M5| + |M4 - M5| + |(M3 ∩ M4) \ (M3 ∩ M4 ∩ M5)|
= |M3| - |M3 ∩ M5| + |M4| - |M4 ∩ M5| + |M3 ∩ M4| - |M3 ∩ M4 ∩ M5|

[dreamy lichen]

but when I plugged in the numbers, it yielded wrong result

[modern sapphire]

yeah

if you see your triple intersection is not cancelled

it gets added x

last term needs to be +ve

- |M3 ∩ M5| removes it once, - |M4 ∩ M5| does it another time

so you need it back 2x

• |M3 ∩ M4| - |M3 ∩ M4 ∩ M5| but this puts it in and takes it away

[dreamy lichen]

why doesnt my approach work too though?

|M3 - M5| + |M4 - M5| + |(M3 ∩ M4) \ (M3 ∩ M4 ∩ M5)|

why is this not true

[modern sapphire]

you discount it 3 times and count it in only once

the parity has to be odd so it stays away

[dreamy lichen]

But this area

• this area

- this area

should yield the intended result

[hot wren]

or not and

multiples of 3 or 4

that would just be 3 and 4

[dreamy lichen]

what?

[modern sapphire]

you are counting the size of the set

so if you repeat the size only once, it doesnt give right answer

your operations are COUNTING the elements, where as unions and intersections dont consider counting the elements

[dreamy lichen]

the magnitude counts the elements though

|M3 - M5| + |M4 - M5| + |(M3 ∩ M4) - (M3 ∩ M4 ∩ M5)|
Would this be a proper way to calculate it? If not, why?

[dreamy lichen]

|M3-M5| gives cardinality of this

[dreamy lichen]

|M4 - M5| gives cardinality of this

[dreamy lichen]

and ||(M3 ∩ M4) - (M3 ∩ M4 ∩ M5)| gives cardinality of this

[dreamy lichen]

where exactly did I go wrong? Does this yield wrong result? Or have I simplified it incorrectly?

Or even calculated it incorrectly 🤔

[dreamy lichen]

This is just inclusion-exclusion principle on sets (M3 - M5), (M4-M5) I think

so it should be correct

[hot wren]

Looks good to me. But you got the right numbers?

the set part looks good

[dreamy lichen]

alright

the numbers are:

Wait does this also look good?
= |M3| - |M3 ∩ M5| + |M4| - |M4 ∩ M5| + |M3 ∩ M4| - |M3 ∩ M4 ∩ M5|

[hot wren]

wait

[hot wren]

|M3 - M5| + |M4 - M5| - |(M3 ∩ M4) - (M3 ∩ M4 ∩ M5)| I think. Since the |(M3 ∩ M4) - (M3 ∩ M4 ∩ M5)| is counted twice

[dreamy lichen]

that's a good point

i think I understand now

thats a really silly mistake

i knew it had to be -

but I wrote +

multiple times

tysm

[hot wren]

yeah hard to spot lol no worries

[dreamy lichen]

ill just quickly check if it works now

,calc (667 - 133) + (500 - 100) - (166 - 33)

[grand pond]

Result:

801

[dreamy lichen]

yes...

it works

tysm

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[foggy tartan]

Why is this wrong? i am trying to proof for a volume of sphere

[slender walrus]

,rotate

[slender walrus]

you shouldn't have the same variable in the bounds and the differential,
and the way you're setting up the rest of the integral doesn't represent the volume of a sphere

assuming you intend
$$\int_{-r}^{r} \pi x^2 \dd{x}$$

[grand pond]

ℝαμΩℕωⅤ

[slender walrus]

using r=2 for example

you'd be rotating

which gives a double cone

with radius r and height r

[foggy tartan]

i am not sure if i undesratnd why you get a cone

[modern sapphire]

in your 3d space, r is not increasing linearly

so your integral in the strictest sense is not in real euclidean space

[slender walrus]

do you have a triangular object?

part of a set square from earlier geometry?

[modern sapphire]

so, your volume of the sliver that you have taken is not really pi r^2 dr since it is somewhat like a cone frustrum

as such your integral formulates a cone

yea

[foggy tartan]

how does equation for an area of circle creates a cone

[slender walrus]

i suggest looking up a vid on solid of revolution, sphere

you'll get visual aids to go with the explanation

[modern sapphire]

integrate pi r^2 dx with r changing as a function of x

which it does in 3d space

but mostly ppl integrate wrt the radius and the integrate over the volume of this spherical shells since the shells form in direction that is normal to the radius vector

[midnight plank]

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[meager galleon]

Need help Explainiing

[fallen aurora]

in what?

[meager galleon]

In the part where it Says
x+y+x-y/(x-y)(x-y) : x^2+2xy+y-x^2/x^2+2xy+y^2-2xy-2y^2

[modern sapphire]

thats just (a+b)^2

do it in latex commands so it changes

[fallen aurora]

either way

should have told me the line

[meager galleon]

?

[fallen aurora]

you mean how they reciprocated it right?

[meager galleon]

Yes

[fallen aurora]

Notice the ratio symbol between two expressions

[meager galleon]

Yes

[fallen aurora]

then what's the matter?

[meager galleon]

Oh I get it

[fallen aurora]

kk

[meager galleon]

Thanks @fallen aurora

.close

[midnight plank]

Channel closed

Closed by @meager galleon

Use .reopen if this was a mistake.

[midnight plank]

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[modern shard]

what have you tried till yet?

[carmine sigil]

I'm not even sure what the question is asking

Is the question concerning a power tower of 5s 24 high?

I.e. 5 ↑↑ 24?

[modern shard]

5^5^5^....22 more times

we have to find remainder

[carmine sigil]

Seems reasonable!

@late palm what have you tried?

!show

[midnight plank]

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

@late palm Has your question been resolved?

[idle phoenix]

i guess no matter what power 5 has the remainder will remain the same

wait no

im wrong

try to find the remainder for 5

then 5^5

therell be a pattern

[modern shard]

I got a better way

[idle phoenix]

k

the answer should be 5 right?

[modern shard]

ye

[late palm]

Yup

[late palm]

Don't know where to start

[late palm]

Pls tell

[modern shard]

what is ending digit of 5^to power of something?

[late palm]

5

[carmine sigil]

So we have a problem mod 24. So every time you get a number bigger than 24 you can replace it with its reduction mod 24

[late palm]

Ya

[carmine sigil]

Something interesting happens when you consider the value of 5^5 mod 24.

[late palm]

Sorry

Lemme do it

[modern shard]

uhmm I was thinking of other approach

[carmine sigil]

Hint: 5^5 = 5^2 * 5^2 * 5

[late palm]

Ya pls tell

[late palm]

Remainder is 5

[carmine sigil]

Exactly, so 5^5 is 5 in mod 24

What about 5^(5^5)?

[modern shard]

finish with approach of omnipotententity

[late palm]

Idk

How to solve such a long question

Ig 5??

[carmine sigil]

Well, 5^5 is 5, so everywhere you see 5^5 you can replace it with 5.

So 5^(5^5) = 5^5

In mod 24

[late palm]

Oh

[carmine sigil]

So...

[jaunty canopy]

that's only true cus 5^24 = 1 mod 24

most of the time this isn't true

[late palm]

So it's 1?

[carmine sigil]

How did you get 1?

[modern shard]

[late palm]

Sorry it's 5

I got a bit confused

[late palm]

5**

[carmine sigil]

@modern shard interested to see your approach.

[late palm]

Ya

[carmine sigil]

I think you'll find 5^24 = 0 mod 5.

[jaunty canopy]

oh sorry i meant mod 24 lol

[modern shard]

well we know 5^5^5 will always end with 5 (an odd number)

general form of odd number= 2n+1

so it will be
5^(2n+1) = 5(25ⁿ) and we can rewrite 25 = (24+1)
which gives
5[24+1]ⁿ
whatever the expansion be, every term would be divisible by 24 except 1 (binomial)
so it would be

24k + 5

hence remainder would be 5

[carmine sigil]

Also a good approach

! occupied

[midnight plank]

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).

[late palm]

Thanks @modern shard @carmine sigil for your help

Also @jaunty canopy

[jaunty canopy]

i mean i didn't really do anything lol

just wanted to point out that if we're trying to find the remainder of a power tower, be careful replacing the exponent

[late palm]

.close

[midnight plank]

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[visual tiger]

!status

[midnight plank]

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

[untold jackal]

1

I am confused on where to start

[modern shard]

,, v_p(n!)=\sum_{k=1}^\infty \left\lfloor\frac{n}{p^k}\right\rfloor

[grand pond]

Vansh

[untold jackal]

cause 1000! is a larger number

[modern shard]

n = 1000, p=7 (btw)

[visual tiger]

just know that 1000! = 1000 * 999 * 998 * ...

so you just have to look at how many times can each of the numbers in the product be divided by 7

[untold jackal]

right

then 7^2

then 7^3

[visual tiger]

yeah

[untold jackal]

and add up all those pairs

[visual tiger]

maybe it's easier to search :

how many are divisible by 7

[untold jackal]

cant the internet cant break down 1000!

[visual tiger]

(so how many multiples of 7 are below 1000)

[untold jackal]

142

as 7 times 142 is 994

and 20 factors of 7^2

and 2 of7^3

is the answer just 166 than

[modern shard]

164*

[untold jackal]

i meant 164 sorry

[visual tiger]

yes

so just to recap

you computed how many multiples of 7 below 1000

[untold jackal]

find how many factors of 7 is bellow a 1000 right

[visual tiger]

(that's floor(1000/7))

[untold jackal]

than 7^2 and 7^3

[visual tiger]

and then how many multiples of 7^2 below 1000

floor(1000/7^2)

etc

[visual tiger]

so this formula

[untold jackal]

whats that formula called

[visual tiger]

doesn't have a name

[untold jackal]

cause it was very usefull to me

[visual tiger]

it's just "p-adic valuation of n!''

[untold jackal]

ooh ok

well thankyou for all of the help it was much appreciated

didnt know something so complicated could just be easy

[visual tiger]

https://en.wikipedia.org/wiki/Legendre's_formula

ig it's also called legendre formula

[untold jackal]

alright well thankyou still for the help

[midnight plank]

@untold jackal Has your question been resolved?

Channel closed

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Use .reopen if this was a mistake.

[untold jackal]

.close

[midnight plank]

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• After 15 minutes, feel free to ping <@&286206848099549185>.
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• 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.

[naive onyx]

how can i tell that the -1 here is power or inverse?

[zealous schooner]

i've never seen that notation used for representing the reciprocal

it's always inverse

although yeah that's not great notation

[naive onyx]

oh oke

yea its most likely it is

cause later in the question it say f(0)

thx

.closed

.close