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[hybrid vine]

What do they mean by represented by a homogeneous equation

[lyric hawk]

a homogeneous equation is one in which all terms have the same degree

in this case, it is 5

[hybrid vine]

I see, if that is every true, I can use the results they have used

i.e the first derivative is of the form y/x?

[lyric hawk]

yeah

[hybrid vine]

All right, thanks

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[random niche]

How would I write up this differential equation in standard form?
$2y' = e^{\frac{x}{2}} + y$

[solid kiln]

Tomi

[random niche]

$y' = \frac{e^{\frac{x}{2}} + y}{2}$

[solid kiln]

Tomi

[random niche]

so im trying to write it up as

[nova spire]

how about putting the "y" term on the left hand side

[random niche]

just take it away from both sides?

[nova spire]

uh huh

[random niche]

$y' - \frac{y}{2} = \frac{e^{\frac{x}{2}}}{2}$

[solid kiln]

Tomi

[random niche]

is this correct?

but you need P(x)y

[nova spire]

yes

so write -y/2 as ...*y

[random niche]

1/2 * y

so P(x) is \frac{1}{2}?

[nova spire]

no

-y/2 is not 1/2 * y

[random niche]

i doont understand the notatoin that you are given me here

well -1/2

sory

[nova spire]

yes

P(x) = -1/2

and now it fits

[random niche]

so v(x) is -\frac{1}{2}x

<@&286206848099549185>

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[radiant portal]

I wrote something in my notes a while ago but I dont understand it now

the graph of a function is a curve too! consider the graph of a function f with x ranging from a to b. This curve is paramitrized by:
integral of x(t) = t
integral of y(t) = f(t)
with a <= t <= b

[radiant portal]

I get what x(t) = t means, but what does y(t) = f(t) mean? What is f(t)?

Btw this is supposed to just be a general thing, not a specific question

Theres also an image that looks like this next to it

[nimble stone]

you could do that with any function y=f(x)
you can say x(t)=t and so then y(t)=f(t), since x=t
i think thats what its saying

[radiant portal]

But what actually is f(t)

Can you give me an example with numbers or something pls

[nimble stone]

if i had y=x^2
i can say x(t)=t so then y(t)=t^2=f(t)

you had y=f(x) any random function, then it just became f(t) since x=t

[radiant portal]

Ahhhhh

I get it now

Thanks the example really helped

so if i had y=4x + 2
i can say x(t)=t and y(t)=4t + 2

Right?

[nimble stone]

its a valid parametrisation yeah

probably the simplest one

[radiant portal]

Ok yeye I get it now

Thank you so much! you explained it really well

❤️

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[oblique plaza]

how should i solve this question

[oblique plaza]

is ae parallel to dc?

[verbal stream]

yes

[oblique plaza]

ohhh

[oblique plaza]

but its not mentioned

[verbal stream]

It's labeled on the diagram

[merry scaffold]

it is drawn"

with this >>

[oblique plaza]

2 arrows mean parallel?

[frail heron]

yes

[merry scaffold]

ye its like an universal sign

[oblique plaza]

oh

got it

ty

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[odd siren]

Hi I need help with this question.

[odd siren]

I need to leave the answers in simplest form and express it in terms of tan theta

i know sin^2 theta / cos^2 theta = tan^2 theta

but idk what to do with the square root and 1-

[pearl lotus]

Sin²x + cos²x = 1 use this

[royal fossil]

$sin^{2} x + cos^{2}x = 1$ use this and substitute the values in for b and take lcm for c and you'll see the answer

damn that's messed up

[odd siren]

haha its ok

ok lemme do this rq

[solid kiln]

furyolen

[royal fossil]

cool, there ya go

[odd siren]

ok im like stupid i still need help

[odd siren]

how do I use this in the q for b

[royal fossil]

did you take the lcm?

[odd siren]

i just did this

[royal fossil]

that is the answer

you said it had to be expressed in terms of tan theta right?

[odd siren]

yeah

I just want to know how i could use sin^2x + cos^2x = 1

to solve this q

[pearl lotus]

[pearl lotus]

Imagine using it and then asking how can u use it

[royal fossil]

taking lcm would give you $ \frac{1 - sin^{2} x}{sin^{2}x} $ with which you can use the above identity since $sin^{2}x = 1- cos^{2}x$

holy cow it actually worked

wait I wrote the damn function wrong

[solid kiln]

furyolen

[royal fossil]

there you go finally

[royal fossil]

man that was a solid jumpscare

[pearl lotus]

Her solving the que without knowing she solved the que was bigger jumpscare

[royal fossil]

real

wait he's a she?

I'll dip before this gets controversial

[odd siren]

im a guy 💀

[royal fossil]

[pearl lotus]

Fck nvm

Sorry sorry by mistake

[odd siren]

its ok

[true kestrel]

[odd siren]

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[subtle ether]

How do I draw the circle for part b

The radius is 13

the centre is negative 2

so to the right of the circle

it should be hitting 11 right

and to the left

it should be hitting -15

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[subtle ether]

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[subtle ether]

I'm not quire sure

how to work out the y coordinates

in part b

of P and Q

IM JUST going

to assume

its tangent line equation

and set x equal to 0

to find out y value

that dont work

fuck

well apparenrtly

one of my y values is correct

I got y = 35

but If I do the equation of the tangent for the other one

it won;t give me the right answer

ahhhh

silly mistake

forgot neg reciprocal

my b

Ok I have no idea how did they get 23

because I have 25

more silly mistakes

crap

why does the distance formula not work

on it

They just do this

but why doesnt the distance formula work

[split chasm]

wdym distance formula doesn't work

if you want to apply distance formula you can and it will work

calculations can just be a lot simpler done the points are on the y-axis

[subtle ether]

so I have the coordinates of (0,35) and (0,23)

If I do root 35^2 +23^2

I dont get 58

also how am I to know that k is less than 0

is a solution

in part b

[split chasm]

why are you squaring both 35 and 23

show exactly how you're applying the distance formula

[subtle ether]

ah i forgot the distance formula

[subtle ether]

But I just plug in the y values

into this

and i get 12

[split chasm]

show how you're plugging

firstly what were your y-intercepts

[subtle ether]

Just plugging in those values

[split chasm]

your second intercept seems to be wrong

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@subtle ether Has your question been resolved?

[subtle ether]

says here its right

ohjhh

its negative

damn

ok thanks

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[subtle ether]

How do I work the coordinates of P

AHHHHHH

FUCKING SIMULTANEOUs

ok

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[subtle ether]

trapezium rule has height/ 2 so it should be -1/12 pi divided by 2 right?

Idk why the mark scheme says 1/12

OHHHHHHHHH

I GET IT NOW

fak

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[drifting storm]

hi guys pls guide me on how to do this

[drifting storm]

the answer is -3<x<5/2

[sacred fern]

2x^2 + x -15<0

[wind cloak]

try to factorise it then

[sacred fern]

Since the top is positive you need the bottom to be negative as we need that fraction to be negative

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@drifting storm Has your question been resolved?

[drifting storm]

oohh

[drifting storm]

(2x-5)(x+3)

btww sorry for the late reply

[wind cloak]

now you know how to solve for inequalities?

[true kestrel]

if you know, you can do like

(2x-5)(x+3) < 0

logic:
since the numerator 5 is always positive, it wouldn't contribute to the sign change hence it only depends upon denominator and since it's denominator it cannot be 0

for solving inequalities you can use wavy curve method

[drifting storm]

oooooo

ohhhh i got it alreadyyy

thank you so muchhh u guysss ☺️☺️

[true kestrel]

thank 77square

i just came in XD

[drifting storm]

yess

thank you @sacred fern @wind cloak!! 😁

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[barren crescent]

do my steps make sense and lead to the correct answer?

[barren crescent]

sorry for the lazy typing and explanation on some its just practice so i was being a little lazy

[wise dagger]

i have doubts on step 7

[dull temple]

I don't get how 4 goes to 5

$\f1{SC}$ becomes $\f SC$ somehow?

[solid kiln]

hayley 🥥 🌴

[barren crescent]

oh

yeah

cosx/(sinx*cosx)

[barren crescent]

wait i dont get why

[wise dagger]

ohhh i get what's wrong

sin²(x)+cos²(x)=1
but NOT
sin(x)+cos(x)=1

[night bay]

I think the 5th step refers to the second term in LHS(the one after negative sign)

[wise dagger]

same for the others

[barren crescent]

this i understand how i messed up oopsie that is kind of an important typo i should have explained what @night bay is saying

[barren crescent]

the others?

[barren crescent]

what does you mean?

[wise dagger]

hmmm maybe i overthought about that

anyways

Why $\frac{1}{\sin x \cos x}$ becomes $\frac{\cos x}{\sin x \cos x}$?

[solid kiln]

Biscuity

[wise dagger]

that's for number 7

[barren crescent]

i didnt explain it well

like my mistake in 5

let me try that again

[bronze ibex]

dude

this is ez

see

[wise dagger]

hmmm

[bronze ibex]

take lcm

you'll understand urself

[wise dagger]

in my opinion, I'd write it out regularly like
L.H.S. = ...
= ...
= ...
first, and then write the explanation on the right side of each step

[bronze ibex]

no

take lcm

so this becomes

cosx.secx - sin^2x/cosx.sinx

[bronze ibex]

cosx.secx = 1

[shut abyss]

Is this where we all getting confused at?

[bronze ibex]

yes this

[night bay]

5th and 6th step seems unrequired and there's mistake in 8th step

[night bay]

yea

[shut abyss]

True, converting into tan is kinda useless

[night bay]

yup

[shut abyss]

Things should be all clear now

[barren crescent]

i should just start from scratch i think

is what you pointed out it or is 7 messed up too

[night bay]

yea that too

[night bay]

till 4th step all is correct

[shut abyss]

5th is also correct, just makes no sense

It's as useless as writing 6/2, instead of 3

[barren crescent]

secx = 1/cosx
so the lhs is (1/cosx)/(sinx) - (sinx)/(cosx)
with that (1/cosx)/ (sinx) becomes (1/cosx) * (1/sinx)
simplifies to 1/(sinx*cosx)

ok steps until 4

[shut abyss]

Yes all correct till now

From now, refer to the pic I send

All confusions will be cleared

[shut abyss]

This

Just don't convert that sin x /cos x into tan x

[barren crescent]

so 5 i need to find a common denominator LHS = 1/(sin(x)*cos(x)) - (sin^2(x))/(sin(x)*cos(x))

[night bay]

correct

[barren crescent]

okay thats something

[barren crescent]

then i need to combine the fractions

lhs = (1 - sin^2(x)) / (sin(x)*cos(x)) ?

[night bay]

yea

[barren crescent]

!!

next pythag identity

(sin^2(x) + cos^2(x) = 1):so my lhs = (cos^2(x)) / (sin(x)*cos(x))

that check out?

[night bay]

yup

[barren crescent]

okay then. i just need to simplify

[night bay]

yess

[barren crescent]

cancel out common factor: lhs = cos(x) / sin(x)

[night bay]

[barren crescent]

oh and i need to see if it = cotx

[shut abyss]

cosx/sinx is the cotx

[barren crescent]

yeep

okay how's the amended version

[shut abyss]

lol, you ain't got book?

[night bay]

yea precise and efficient noice

[barren crescent]

i aint got book?

if you mean an answer key

[shut abyss]

I mean it's easier to write

[barren crescent]

no

oh

yeahhhh

i have eds

so my hands tend to overextend when i right and it causes injury

annoying asf to type it out but doesnt hurt

[shut abyss]

Oh

[barren crescent]

not tryna medical info dump i wish i could use book all the time too lmao

[shut abyss]

Anyways, you got all clear

[barren crescent]

yipee!

thank you all!

[shut abyss]

np

[barren crescent]

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[barren crescent]

what mistake did he make??

[barren crescent]

idk im just not connecting the dots

think the right answers are pi/6, 5pi/6, 7pi/6, and 11pi/6

[strange stag]

he didn't consider -1/2

that's his mistake

[barren crescent]

-1/2? in what equation?

[strange stag]

when square rooting the 1/4

he didn't get 7pi/6 and 11pi/6 for this reason

[vital tusk]

are you guys in summer school 🤓

[strange stag]

what is summer school???

[vital tusk]

idiots

[trail ingot]

if sin^2(theta) = 1/4, sin(theta) could be 1/2 or -1/2

[strange stag]

is this some american thing or what

[vital tusk]

you guys wanna see my equation i made

[barren crescent]

no I'm doing the course online and the semesters don't match regular public school

[vital tusk]

?

[barren crescent]

it's school in the summer not summer school

[barren crescent]

summer school is remedial accelerated coursework to recover credits in the summer for students who didn't get the required credits during the school year usually due to failing classes

[strange stag]

oh, credits isn't a thing in my country-

[barren crescent]

unfortunately it is here!

[barren crescent]

one last question, do these look good?

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[wraith hinge]

How would I integrate to find the volume of a shape like this:

[wraith hinge]

Using multiple integrals

These are the full equations:

The part I'm confused about is that if I try to setup the integral with the cross section in the zy plane from x = 0 to x = 7.5 it is not integratable because there will be an x in the end which cannot be integrated with respect to y.

I don't think there is a change of variables either because the x y plane and z x plane don't have exact cross sections

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[wraith hinge]

<@&286206848099549185>

So the integral I found to evaluate the volume looks like this:

I just need help evaluating this integral

[bold echo]

Don’t know if I can help, however the integral you found isn’t showing up for me

[wraith hinge]

actually that is an integral for volume of half a wing

but its pretty much the same thing

[bold echo]

Is it that the integral most inside has a range of 0 to 7.5 and the others are only larger than 0?

[wraith hinge]

I'm not sure what that means

[bold echo]

Like the numbers on the integral sign. Am I missing something, since the integral sign closest to the function has numbers 0 and 7.5. The other integral signs only have the number 0 on the bottom, so is that how it’s supposed to be or did I not see something?

[wraith hinge]

Yeah the numbers at the bottom of the integral signs are all 0s

[bold echo]

So what are the numbers on the top? I see one as 7.5.

[wraith hinge]

the upper bounds for integrals are 5-(1/2.5)x and -y(y/c(x)-1) for the first two integrals

[bold echo]

@frigid rampart

[wraith hinge]

Can they help?

[frigid rampart]

I'm thinking about it. It's been a while.

[bold echo]

Yeah, pretty sure they can

[frigid rampart]

You want to do the dz integral first, dy second, and dx third.

The dz will have the upper and lower bounds of the z_top and z_bottom functions.

The bounds of the dy integral are 0 and the line

y = c(x)

[wraith hinge]

the thing is I don't see the projection in the xy plane, its just lines so I don't think I can change the order of integration

[frigid rampart]

I would do this in two parts:

1. Only do the right half of the wing, since it is symmetrical.
2. Do the top and bottom parts separately.

The cross-section in the xy plane is a trapezoid, x from 0-L, y from 0 to c(x).

The last integral needs to have the simplest bounds.
The first integral can only have simple bounds if the other variables do not depend on it.

[wraith hinge]

Is the right half of the wing the part that is a rectangle without the triangle part

[frigid rampart]

The right half is the right half of the two parabolas. Using Figure 1.

c(x) is y_left -> y_right, so use c(x) for y_right and y_left = 0.

Though that might not matter, the parabola will become a cubic and it will work out the same.

[wraith hinge]

how did you find that the cross sections in the xy plane were trapezoids?

[frigid rampart]

Because
x : [0, L]
and
y : [0, c(x)]
where c(x) is a straight line. I saw the straight line in the Desmos graph.

Probably skip what I said about splitting it right and left, Just go with
y : [0, c(x)]
as given.

[wraith hinge]

Okay so just cutting it like that where the two parabolas meet and integrating?

[frigid rampart]

Exactly. Because the wing is cut off where the two parabolas meet.

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[wraith hinge]

Would this formula also work on surface area? Can I keep the bounds when I set up the surface integral and solve for a surface integral normally?

[wraith hinge]

@frigid rampart

[frigid rampart]

The surface area integral is completely different. It is 2-D, dx dy, so the surface element comes from the tilt of the wing surface. If the surface is flat,
dx dy ==> (x2 - x1) * (y2 - y1)
but when it is tilted, it becomes bigger.

[wraith hinge]

Aren't we talking about this formula?

Why can't I just plug in the bounds for y and x that we found earlier into the double integral and use z top and z bottom to solve for the inside square root function?

To solve for total surface area of the shape

The formula I am using:

<@&286206848099549185>

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[wraith hinge]

close

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[frail grotto]

<@&286206848099549185>

[knotty quest]

!15m

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[knotty quest]

And yea what's the problem?

[frail grotto]

can you complete this limit?

[knotty quest]

What have u tried so far?

I shouldn't be giving u answers

I should be helping only

[frail grotto]

yess I'm confused

[knotty quest]

Mhm

Which one?

[frail grotto]

I studied on the internet and didn't understand

[knotty quest]

Mhm

Ok so for the first one

[frail grotto]

pleasee

[knotty quest]

Factorise numerator and denominator

Ull get

[frail grotto]

i don't knowww😫😫

[knotty quest]

What do u not know?

Factorising?

[frail grotto]

aahhh i seee

thank you

[knotty quest]

Anytime!

[bronze ibex]

what are u stuck on?

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[sterile egret]

In solving integral of
x^m .(ax^n+b)^p type when p is not an integer, we check that (m+1)/n is an integer or not why so

[sterile egret]

?

What's the logic behind those substitutions?

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[loud orbit]

I just need a check on these if someone can tell me if I got them right or not

[ancient edge]

Looks good to me

[loud orbit]

Alright thank you

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[grizzled flower]

Why did we look for R like this instead of just multiplying the vector of AB by 1/2 to get the middle of it?

[grizzled flower]

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[wide sapphire]

hi! i need someone to check to see if i got this question right. (:

[orchid wagon]

you could just replug the solutions into the equation and see if it is true

[wide sapphire]

i know, i just need validation from another professional

[orchid wagon]

looks correct

[wide sapphire]

thank you

[orchid wagon]

[limpid dawn]

💀

[wide sapphire]

what abt these

[wide sapphire]

oopsie

[limpid dawn]

it's aight

[wide sapphire]

mhm

i just need a check mark or an x

[limpid dawn]

i am sorry it's just you posted so much

10 in 1

[wide sapphire]

theres a person in the server idk what their user is, but they just draw a check or x, shoutout to that person

idk how he/she does it 💀

but it turns out to be correct 📜

i just pray the person is online

WHEREVER YOU ARE

[limpid belfry]

i could try to help

[wide sapphire]

please 😭

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[frigid kiln]

second part? i got 80.9 deg as one answer and the ms says its correct but it also says i have to perform another method to get 211.7 deg. i have no idea what i have to do

for the first part the answer is 2 sqrt 13=R and alpha=56.31 deg

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@frigid kiln Has your question been resolved?

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@frigid kiln Has your question been resolved?

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@frigid kiln Has your question been resolved?

[frail heron]

Do you kno9w why there is an another answer?

This is the first info you need to know

[frigid kiln]

No i dont

[sleek basin]

because sin25 °=sim385 °

so there will be another answer

sin(360 °+X)=sinX

[wind cloak]

it will be sin(155degress)

[sleek basin]

@frigid kiln do you understand

[wind cloak]

theta is between 0 and 360 degrees

[sleek basin]

same meaning

[wind cloak]

no😶

[sleek basin]

need to know the meaning of sin

[sleek basin]

=sin155deg

=sin25deg

[wind cloak]

yeah

[sleek basin]

=sin385 °

[wind cloak]

but this thing is not correct here

theta is less than 360 degrees

[sleek basin]

i’m explaining the conception

[frail heron]

He’s elaborating that the right side is equivalent to the left one.

[sleek basin]

yes

[wind cloak]

okok i got what he means now

[frail heron]

Good, enjoy your stay

[sleek basin]

so you find a degree that confirm the answer will be multiple if there is no limitation

but it gives you maximum to 360 will be 2 deg

[frigid kiln]

Ou yes i get that its like but when i did 180- 81 from the cast diagram its not the answer

Is that what u meant

[frail heron]

Could you share your process with us?

[sleek basin]

yeah that will be easy to realize

[frigid kiln]

Okay

1 sec

I hope that includes the main working i scribbled out like a page of wrong stuff😞

I did the sin part of the cast diagram because i took the inverse of sin. The tan was just there for accuracy but it was already calculated before

[frail heron]

@frigid kiln

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[frail heron]

.reopen

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✅

[frigid kiln]

I dont get it where did tan B come from, was there a triangle i was supposed to use?

Like in the previous answer i had a right angled triangle to find R

[frail heron]

A and B are the angles I assumed

[frigid kiln]

Idk if that has anything to do with it

[frail heron]

The graph I provided demonstrates how the second answer appears

sinA can either resides in the first or the second gradient

Yielding different values of the angle

[frigid kiln]

sin A can be in the first and second quadrant while tan can be in the first and 4th quadrant right, is taht what ur saying? also im so sorry but why r u using tan? if we want two angles cant we just use the 2 sin quadrants?

[frail heron]

Third, not the fourth

[frigid kiln]

oh mb

yeah third

[frail heron]

Furthermore, that’s what you assumed in your previous equation

Theta = sin^-1 A + tan^-1 B

[frigid kiln]

yes so r u saying because i had a tan there thats why the use of tan emerges? or did i not get u

[frail heron]

yeah, you used tangent in your equation

That’s why I used tangent in my graph

[frigid kiln]

ok now ig i understood why we used tan, but now why did ur tan only have 3/2 while mine had smth else

[frail heron]

6/4≠3/2？

[frigid kiln]

oh lol yes it is equal to that mb, lemme retry that part of the question o see if i see what ur really getting at 1 sec

[frail heron]

Certainly

Ping me when you’re done

[frigid kiln]

BRO IM SO SORRY but im lost.

what i know rn is we have to use tan because i used a tan in my equation, i have assumed that there is more than 1 answer because there are multiple quadrants

Now i dont get how im supposed to use tan at all. Pregiously my only use of sin was because i substituted my values of R and alpha into the equation Rsin( theta - alpha ), then i took sin inverse because i was bringing it to the other side in krder to isolate the theta. I get that i have to use tan but HOW do i use it if the original form only used sin? what do i tan?

Even if i use tans quadrants on my answer 180+ 80.9 is not correct

Small update
Ive been trying harder and looked at ur diagram for reference only the sin moved so a theta value for sin A from another quadrant and added THAT to my tan^-1( 6/4 ) and got the 211.7 honestly but even if thats correct i dont get it it all seems so puzzly

Is there a reason why only sin moved while tan ddint? Or did i assume the stuff wrong and we never had to really change tan only sin

Sorry if this is confusing

[frail heron]

Bruh, you don’t need to apologize for being confused

[frigid kiln]

yeah but my confusion might start making u confused

[frail heron]

First of all, either using sine, cosine, or tangent are ok

It doesn’t matter at all

You just need to use those to represent the angles

[frigid kiln]

using them where? in the Rsin(theta-alpha) equation?

[frail heron]

Look at the first ss

You moved 2sqrt13 from the left to the right

And look at the second, you move sin to the right side, yielding sin-1

What I wanted to notify you is that which trig you used doesn’t change the angle

[frigid kiln]

okay i get that so for example in a triangle cos^-1(Adj/hyp) would give the same as sin^-1(opp/hyp) right the angle itself isnt changing theta is still the same

only the trig is changing

thats what ur saying right

[frail heron]

When transforming sinB from tanB, only the ultimate value would changed

[frigid kiln]

the ultimate value is the value inside the trig function right? so sin (x) the x would change

or is it the value its equal to so sin(X)=y so y would change

[frail heron]

For example
sinX = 5
TanX ≠5

[frigid kiln]

so from this x isnt changing and normally x is theta, theta doesnt change only the output of the trig function on it

[frail heron]

yep

[frigid kiln]

so how do i apply that concept to problems like these?

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[dusky tinsel]

can anyone pleae help me with how to do this question, how do u find a) or lambda?

[gilded pulsar]

Start by finding a general formula for the area depending on x and y

[dusky tinsel]

hm

[dusky tinsel]

i can find the maximum and x intercepts

but idk how to relate it to the thing

+- sqrt(49/3)

[gilded pulsar]

im sorry what exactly is your question

[dusky tinsel]

i cant find the general formula for area depending on x and y

can you help

[gilded pulsar]

now what is the general formula of the area of a triangle

[dusky tinsel]

1/2bh

or 1/2absinc

i mean 0.5 * 2x * y = Area

so 0.5 * 2x * 49 -3x^2 = A

[gilded pulsar]

if you dont forget the parenthesis

[dusky tinsel]

+- 7/3 = x

[dusky tinsel]

so which ewquaiton do i sub it back into, the 7/3

i can do 0.5 * 2(7/3) * y = A

[gilded pulsar]

your area equation

[dusky tinsel]

one issue

[dusky tinsel]

how can we sub y = 49 - 3x^2 into the area equation

if its not actually 'y'

its a random variable like a, b ,c etc, or is it not

[gilded pulsar]

why is it not y

[dusky tinsel]

well y is the curve right

but we set y as a line there

asw

arnt they 2 different things

[gilded pulsar]

no y is the y coordinate of the curve at a certain x coordinate

[dusky tinsel]

oh tru

yea lol

[gilded pulsar]

the curve then is all points (x,y)

that satisfy the condition

[dusky tinsel]

ah yep

i go tit

thanks

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[safe solstice]

.

a bookshop has 10 novels and 8 biographies

[safe solstice]

how many ways can I buy 2 novels and 3 biographies

[marsh forum]

!status

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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.
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[safe solstice]

I know that it's 45 ways you can buy 2 novels and 56 ways you can buy the 3 biographies

but I want to know why it's 45×56 rather than 45+56

that's a good question

okay

it's 6

I get why it's 6

and in theory I should get why the books situation is done in that way

okay yeah I got it ty

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[safe solstice]

.reopen

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✅

[safe solstice]

hold on

i got a different question

wait

same as before

10 novels and 8 biographies

but how do I find how many ways I can buy 4 books and with at most 1 novel

okay can you give me an anology of why it's (3 bio 1 novel) + (4bio 0 novel) and not (3 bio 1 novel) × (4bio 0 novel)

why is it addition in this case

alright I think I got it

ty

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[marsh forum]

what?

What's angle 1?

[zinc raft]

Where it says 56

On the circle

That’s angle 1

[marsh forum]

Cool.

[zinc raft]

Ok nvm…

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[dense wasp]

hello

[dense wasp]

I want to ask about these 2 symbols in mathematics

what's the difference between these 2 symbols? at which situation should I use one of the symbol or the other one?

[hallow kite]

https://math.stackexchange.com/questions/4219128/when-to-use-to-vs-implies-in-discrete-math

[modest spade]

usually the left symbol is a mapping

The right symbol is ‘implies’

[dense wasp]

do u mind extending on that?

[modest spade]

well, some use it like this

[dense wasp]

I just want to make sure I use the right symbol on my proof's solution

[modest spade]

f:A->B

Defining a function f that maps the set A to B

[dense wasp]

alright

that's pretty common

[modest spade]

Very

there are other uses though

[dense wasp]

is there any case/situation where the 'mapping' symbol is used as implications?

[modest spade]

a lot of people confound their notation a lot of the time

It’s about defining your notation

[orchid wagon]

the right means "implies". A => B means that if A is true, then B is also true. this is not reversible

the left can either mean "approach". x -> 0 means "as x approaches 0". or it can also describe the domain and codomain of a function

[modest spade]

Very true, there is another meaning for ->

[modest spade]

But conventionally, you don’t use it as implication

so I’d say don’t but if you insist all that matters is definition and clarity of definition

[dense wasp]

alr

[modest spade]

Essentially

Maths is a canvas

Assert your axioms

And language and you’ll get a painting

[dense wasp]

Well lets say, it is assumed that A is true, and from that assumption, it turned out that B is true based on the assumption, can i say that A=>B is true despite whether or not A is true?

it's like the mathematical induction proof

[orchid wagon]

yes, true => true is true

[orchid wagon]

false => true is also true

[dense wasp]

if A is turned out to be false, will that make A=>B false when if A is assumed to be true, A=>B is true?

[dense wasp]

ah yes

ok

[dense wasp]

frfr

[orchid wagon]

no, the only instance when A => B id false is when A is true and B is false

[dense wasp]

alrite then

[dense wasp]

I see

thank you so much y'all

I'll close this now

.close

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[orchid wagon]

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[hazy viper]

Hi

[rotund grove]

!original

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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.

[rotund grove]

!da2a

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[hazy viper]

is it acceptable to simply like this?

simplify*

[true sierra]

Looks good to me, although with infinities it can be a bit wonky at times

[hazy viper]

The problem is

i get this times 2

this is how its suppose to look like

(this one does not have 1/2 multiplied in)

[hazy viper]

but i have 1/2 x that

but i see nothing wrong with that i did...

[ionic pendant]

try plugging in a few values of n to the formula and see what happens

[hazy viper]

thats what i do

[hazy viper]

i get this divided with 2

because i have this 1/2 factor outside of the sum sign

[ionic pendant]

note that for odd n the formula give x^n - x^n and for even n the formula gives x^n + x^n

[hazy viper]

i know...

[hazy viper]

i get this divided with 2

but i see nothing wrong here

[ionic pendant]

well if it's both multiplied and divided by 2 then that is correct, no?

[hazy viper]

what?

[hazy viper]

this is the correct answer

i get the correct answer divided with 2

[ionic pendant]

well we get x^n - x^n = 0, and x^n + x^n = 2x^n

[hazy viper]

no we dont

we get x^n * (1 + (-1)^n) on the top

so its 0 when n is odd

[ionic pendant]

and it's 2 when n is even

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[hazy viper]

@ionic pendant i understand now thank you

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[glad patio]

how do i do this

[rotund grove]

ok

[clear cloud]

With x^

[glad patio]

so do i just multiply root x

by itself

on the top and the bottom

[cobalt peak]

You could convert 1/sqrt(x) to x^(-1/2) and then use the properties of exponents

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[glad patio]

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[forest mountain]

i need help on the first point pls

[blazing meadow]

the ":" meaning division right?

[forest mountain]

yes

oh wiat

yes

[blazing meadow]

have you started the problem yet?

[forest mountain]

no 😦

[blazing meadow]

try to start by solving piece by piece. for example:

do that for every piece and then try to reduce the expression

[forest mountain]

i think it is 3^15 / 4^15

thats right?¿

=0,0133

[shrewd obsidian]

yeah, but i think you should solve everything algebraically, and in the end you can compute it numerically

just manipulate them symbols

[forest mountain]

.close

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[tame sinew]

A set of vectors { v 1 , v 2 ,..., v k } is linearly dependent if and only if one of the vectors is in the span of the other ones.

I feel like this is a dumb question but

if you construct a set of vectors where two linear combinations from two distinct subsets can create the same vector, does this imply that there's a single vector in the set that's in the span of the other ones?

[hallow kite]

could you give an example of what you mean by two linear combinations from two distinct subsets?

[tame sinew]

sorry if the phrasing is hard to understand

[wraith hinge]

when you say distinct, you really mean distinct not disjoint?

[hallow kite]

technically if you can find two linear combinations of vectors from two distinct subsets that produce the same vector, and those linear combinations contain at least one nonzero coefficient, then you can guarantee that the set is linearly dependent

[tame sinew]

disjoint

[wraith hinge]

if you assume they create the same non-zero vector then yes

[hallow kite]

if i consider the set {v1, ... , vk} and i consider two subsets {v1, ..., vn} and {vn, ..., vk}, and there's some linear combination from the first set that produces a vector vp and another linear combination from the second set that also produces vp, then you can subtract one linear combination from another and find a linear combination of all vectors in {v1, ..., vk} that produces vp - vp = 0, so your set is LD

assuming that v_p is nonzero, as mentioned

technically v_p can be 0 but that would require that at least one of the two subsets is LD

[tame sinew]

I understand that this would make it linearly dependent by definition since there's a non-trivial solution to find Ax = 0, but is there a way to justify that there has to be some single vector that's a linear combination of the others based on this starting point

[hallow kite]

based on the starting point?

if you know that the set is LD then you know that there is a vector in the span of the others

let's say v_p = c_1 v1 + ... + c_n v_n and v_p = c_n+1 v_n+1 + ... + c_k v_k, then you can set them equal to each other and always rearrange for one of the vectors in {v1, ..., vk} in terms of the others

if that's what you mean by the starting point

which technically isn't any different from subtracting one linear combination from the other to get the zero vector for LD

[wraith hinge]

yes, exactly that, because of the nonzero assumption, we can always find a nonzero coefficient

[tame sinew]

sorry for the trouble, I think I phrased the question in an odd way, but basically what I was wondering is:

given {v1, ..., vn} and {vn+1, ..., vk},

if we know there's a set of integers c s.t c_1 v_1 + ... + c_n v_n = c_n+1 v_n+1 + ... + c_k v_k,

can we directly use this property to justify that there's some vector v_i in the union of the two sets that's a linear combination of the other vectors, or do we just infer that there has to be one based on the fact that we know there's a linear dependence

[wraith hinge]

you wrote vn in both sets, do you mean they are not disjoint? But have one common vector in both sets?

[tame sinew]

fixed it, my bad

[wraith hinge]

no you don't need anything other than that there exists 0≠v\in span{v1,...vn}\cap span{vn+1, ..., vk}

[tame sinew]

I see. thanks for the help @wraith hinge and @hallow kite.

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[graceful willow]

is this correct?

[graceful willow]

actually is the answer for a) 634.96

[limpid dawn]

What's the expo function?

[graceful willow]

and b) -9.78

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[graceful willow]

my answer that I revized was correct

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[covert wolf]

What is size of angle x when there's pyeonghaengsabyeonhyeong ABCD and the outer angle of A is angle x

[covert wolf]

Also there are 5 possible answers
65°,70°,75°,80°,85°

It means 평행사변형

[violet gust]

So I translated, it means parallelogram

Maybe a figure might help? there are no numbers in your question

[violet gust]

so, since it is a parallelogram, the angles A + B = 180

and you know A/B ratio is 5/4

so find A, and 180 - A is x

Or B which is same as x

[covert wolf]

The problem just says x:x=5:4

A and B are same angle but it has different ratio, and there's no choice of no answer

[violet gust]

If A and B are same, then the ratio is 1:1, cant be different. That also makes the quadrilateral a special case, i.e. a rectangle

But I am pretty sure the problem says A:B = 5:4 even if I cant read korean

[covert wolf]

a and b are corresponding angles

∠A=∠B=x

Isn't it

[violet gust]

sry

they cant be corresponding angles

the standard notation means, the angles A and B are interior angles of the quadrilateral

so A = the angle BAD and B is the angle ABC

[covert wolf]

I think I solved it

[wraith hinge]

if A:B = 5:4 and 180-A=B, A+B=180, so 5+4 =180
180 = 9
180/9 = 20

hi

im new here

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[tidal nebula]

i knoww angle bcd is 99 but dont know how to find out the variable

[vagrant marsh]

use the fact that AD || BC

[tidal nebula]

oh ok

so theatre is equal to 99?

[hushed stirrup]

what do u know about angles formed by lines crossing parallel lines

[tidal nebula]

sorry i meant 81

[hushed stirrup]

yes

[tidal nebula]

thanks

also stuck on this question

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[earnest nymph]

is there a way to do b without breaking it up into cases?

[earnest nymph]

if theres a more concise method than this, pls let me know

[shrewd ridge]

no, no way

[earnest nymph]

ah

[shrewd ridge]

ok wait

[earnest nymph]

i just have to live with this then?

[shrewd ridge]

9 is one less than 10

[earnest nymph]

ye

[shrewd ridge]

the answer should be the same for a and b

[earnest nymph]

it is, but can you explain why pls

[shrewd ridge]

there's a bijection, every 10 letter word corresponds to one 9 letter word

and vice versa

[earnest nymph]

ahhhhh

i think i need formal proving for this and they prob wont accept this tho

nice idea

thanks for the help

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[limpid fulcrum]

If x^2 + 25y^2 + 4x - 150y + 204 = 0 is an elipse equation. Find its Area

[limpid fulcrum]

so this doesnt look like an elipse equation at all

[exotic pine]

it is an eclipse equation

[velvet coral]

wow, instant boost with my first view in the server, let me find some stuff about it then

[limpid fulcrum]

Its says its an ellipse

[exotic pine]

wait

sorry, yeah uts an elipse ewuation

mixed it up with eclipse, sorry

[limpid fulcrum]

Yeah thing is it doesnt look like an ellipse equation

And i cant find a and b to to find the area

[exotic pine]

uh

what form of an elipse equation do you know of

[limpid fulcrum]

Something like this

And area=ab(pi)

[exotic pine]

er

[velvet coral]

so as far I know

[velvet coral]

yeah this is the equation, general one

[exotic pine]

er i have no clue

[velvet coral]

$$\frac {x^2}{a^2} + \frac {y^2}{b^2} = 1$$

[solid kiln]

Greydawn Dewer

[limpid fulcrum]

yeah so how do i turn it into the general one

[exotic pine]

you can factor the oroginal equation to look something like a(x+n)^2+b(y+m)^2=c

[wraith hinge]

you will have to translate the origin

[velvet coral]

a and b represent major and minor axis's

[velvet coral]

hm

factorization, yeah

[limpid fulcrum]

what do i take as factors

[exotic pine]

and since we are talking about are we dont care about its center so ax^2+by^2=c

[exotic pine]

you can complete the square

[velvet coral]

I got the answer, but I wont tell you rather give you the idea

and I cant use a notebook as this is a PC so

[limpid fulcrum]

okay thing is... During corona we didnt learn how to factorize and make a square equation i only know how to solve one

[exotic pine]

wtf

[velvet coral]

just try to factor in 3 things, 1 - x^2 terms 2 - y^2 terms and 3 - constants

and then just form a equation, then try to make that general form from that equation

[exotic pine]

um if you have some equation x^2+2bx+c you can factor it as (x+b)^2-b^2+c

[limpid fulcrum]

this is my attempt at it

[velvet coral]

lets check

[exotic pine]

you want to make it squares

[limpid fulcrum]

uh

[velvet coral]

try to deal with 204 later

[wraith hinge]

https://en.wikipedia.org/wiki/Completing_the_square

[velvet coral]

deal with the X^2, X and Y^2 and Y terms first

[velvet coral]

yeah

if you forgot, you can CREATE 0 out of now where, for example, 0 = 1 - 1

use this to factor X and Y

[velvet coral]

into sections ofcourse

[exotic pine]

you want to use squares

[limpid fulcrum]

You mean x^2+4?

[velvet coral]

you... dont want to DISINTEGRATE squares

[velvet coral]

yeah

ok, let me show you one step for you to get the idea

[limpid fulcrum]

but the x will go multiply with the 4

[velvet coral]

first group it like

[exotic pine]

here lemme try to give you an example, you want to factor $$x^2+2x-3$$
$$(x^2+2x+1)-4$$
$$(x+1)^2-4$$

[solid kiln]

Skill_Issue

[limpid fulcrum]

what the hell is that

[exotic pine]

algebra

[limpid fulcrum]

How did the -4 seperate out

[velvet coral]

you need that to come here

[velvet coral]

the concept of 0 = 1 - 1

[exotic pine]

-3=1-4

[velvet coral]

exactly

[exotic pine]

the key thing is to look at the 2x

[velvet coral]

lets help him for one step

[limpid fulcrum]

ooookay but how would that help if we made it +1

[velvet coral]

$$x^2 + 4x + 4 - 4$$

[solid kiln]

Greydawn Dewer

[velvet coral]

see?

[limpid fulcrum]

wait

[exotic pine]

(x+a)^2=x^2+2ax+a^2
so you wabt to find a that fits as the number infront of x

[limpid fulcrum]

I think im getting it now

[velvet coral]

make one for Y yourself

[limpid fulcrum]

okay one moment

[velvet coral]

I learnt that in 4th Grade lol

[exotic pine]

wtf fourth??