,w solve 2/x + 3/4 = 0
#help-27
restive river
woven radishBOT
mild bronze
2 * 4 = x * 3
restive river
The magnitude of your answer is correct but the polarity isn’t
restive river
2 * 4 = x * 3
You’re forgetting the negative sign mate
zenith jacinth
IS NEGATIVE
calm down man
zenith jacinth
LITTLE PRICK
welp, you do you
restive river
Be nicer next time if you need help….also no one is forcing you to stay here….you are free to leave
<@&268886789983436800>
restive river
I can’t help you I’m afraid. My apologies.
modest dagger
IM SORRYYYY
You're muted for now. You can come back in a day. Until then please reread the rules.
hardy tartan
You're muted for now. You can come back in a day. Until then please reread the r...
idiot
devout snowBOT
@mild bronze Has your question been resolved?
devout snowBOT
devout snowBOT
Available help channel!
hollow bolt
devout snowBOT
hollow bolt
please explain
devout snowBOT
@hollow bolt Has your question been resolved?
ornate raft
$u_k + \frac{1}{2} = 3(u_{k-1}+\frac{1}{2})$
woven radishBOT
mindovermatter01
ornate raft
$u_k = \frac{5*3^n-3}{6}$
woven radishBOT
mindovermatter01
devout snowBOT
@hollow bolt Has your question been resolved?
remote pond
is this the correct answer to the problem?
devout snowBOT
is this the correct answer to the problem?
flat maple
3x+8=0
3x=-8
x = -8/3
devout snowBOT
@remote pond Has your question been resolved?
devout snowBOT
remote pond
.reopen
devout snowBOT
✅
stray spade
stop using multiple help channels
devout snowBOT
@remote pond Has your question been resolved?
lost gazelle
ive just started dot product, and i have difficulties solving q12, can anyone guide me through it?
devout snowBOT
ive just started dot product, and i have difficulties solving q12, can anyone gu...
lost gazelle
ive tried solving it, but the angle is really getting me
restive river
ive just started dot product, and i have difficulties solving q12, can anyone gu...
can u send a more zoomed image of it?
lost gazelle
restive river
thnx mate
aight so u have the magnitude of a - b vector
now a tip form me is
whenev u see
either
mod of a - b
or a + b
or any linear multiple of these two vector
just square both the sides
cuz
lost gazelle
and i have a•b
restive river
a . b
lost gazelle
tysm dude
devout snowBOT
devout snowBOT
Available help channel!
dawn slate
can someone pls help
devout snowBOT
can someone pls help
devout snowBOT
@dawn slate Has your question been resolved?
dawn halo
i know it isnt really maths, but logics, but i didnt know where to ask but here
devout snowBOT
can someone pls help
dawn slate
<@&286206848099549185>
twin peak
<@&286206848099549185>
In the first column, we observe:
(7×2×3)+(7+2+3)=42+12=181
Similarly, for the second column:
(9×5×6)+(9+5+6)=270+20=410
When applying the same logical structure to the third column, we arrive at:
(13×7×10)+(13+7+10)=910+30=940
Add it up, and what does it come to? @dawn slate
restive river
In the first column, we observe:
(7×2×3)+(7+2+3)=42+12=181
Similarly, for the s...
Wow you figured it out! Impressive
I couldn’t do it
twin peak
I couldn’t do it
Nah heck my mistake, and I'll buy you a burger too lol
fringe loom
In the first column, we observe:
(7×2×3)+(7+2+3)=42+12=181
Similarly, for the s...
sweat
restive river
Nah heck my mistake, and I'll buy you a burger too lol
✨
fringe loom
how the hell did u figure that out
restive river
how the hell did u figure that out
Big brain thinking fr 🫶
Next level pattern recognition
restive river
nvm what the hell is that
What the [insert h word)] is what?
dry geyser
In the first column, we observe:
(7×2×3)+(7+2+3)=42+12=181
Similarly, for the s...
I don’t get the first row. What do you mean by 42+12=181?
dry geyser
Well 42 and 12 don’t add up to 181 at least
devout snowBOT
@dawn slate Has your question been resolved?
twin peak
Well 42 and 12 don’t add up to 181 at least
Yeah lmao
what hell i just do
add up haha
dry geyser
Pattern recognition 📈
Addition 📉
twin peak
Pattern recognition 📈
Addition 📉
In Brazil, we use shorter words instead of the full ones, and sorry haha
dry geyser
It’s fine lol, just a bit funny
sturdy mango
if the position vectors of points 𝐴,b, and C are non-collinear, then the points themselves are also non-collinear? ans says nah but idk how to think about this
trail eagle
devout snowBOT
@sturdy mango Has your question been resolved?
proud scroll
hello, how should one go about solving somehting like this? thank you in advance.
devout snowBOT
hello, how should one go about solving somehting like this? thank you in advance...
acoustic leaf
,rcw
woven radishBOT
knotty sage
hello, how should one go about solving somehting like this? thank you in advance...
logarithms is an inverse operation to exponentiation
proud scroll
logarithms is an inverse operation to exponentiation
i know that but i am dumb it doesnt help me
knotty sage
i know that but i am dumb it doesnt help me
so simply to solve for x^2=1
we take square root of both sides
and to solve sqrt(x)=1
we square both sides
it’s kind of like the same process
to solve for 4^x=y
we take log_4 on both sides
and to solve log_4(y)=x we exponentiate both sides using 4
restive river
Well, the question basically asks:
What is a number raised to the power of a number which is equal to the number that the base number should have to get the uppermost number. And the base number indeed has that number. So the answer would be 5.
sturdy mango
this is quite useful property you should remember
very handy in calculus stuff too sometimes
proud scroll
very handy in calculus stuff too sometimes
we did not get to calculus yet. but apparently we will do it right after this.
knotty sage
we did not get to calculus yet. but apparently we will do it right after this.
still it’s just the same property in your question
sturdy mango
this is quite useful property you should remember
its just something to remember
and you can place some numbers and see why it works too or look up the proof online
dense venture
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solution was wrong.
I know that both converge to e but we haven't discussed series, sequences or convergence yet, only inequalities, proof methods etc.
So I don't know how to prove the convergence process.
Showing it numerically feels wrong.
Thanks in advance : )
lusty sapphire
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
!show
devout snowBOT
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
Show your work, and if possible, explain where you are stuck.
restive river
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
have you discussed fundamental sequences
or limits by any chances...
not really provable without those things..
dense venture
Limits no, not sure what you mean by sequences exactly. But things like the sum sign sure
lusty sapphire
Hope it's fine as pdf
screenshots are usually preferred
dry geyser
It looks like you assumed that a < b and a < c somehow implies b < c (the red arrow on the second page)
dense venture
It looks like you assumed that a < b and a < c somehow implies b < c (the red ar...
Yes, since only the denominator was changed (increased) in the pink highlighted I thought it is fine to assume that its smaller
Is there another way to convert the formula
dry geyser
If you can show that the sum from i to n divided by n+1 is smaller or equal to 1/(n+1)! then you can prove the inequality quite straightforwardly I think
But idk how easy that part is
Actually it’s just wrong I think
So impossible to prove
Hmm
Yeah it’s the same thing you did after all lol
Maybe induction is the wrong approach here
dense venture
Had the idea as well, but couldn't come up with a better plan
lusty sapphire
Maybe induction is the wrong approach here
If true, quite counterintuitive for sums
dry geyser
If true, quite counterintuitive for sums
Yeah it feels like induction must be right. But I don’t see the way yet
lusty sapphire
@dense venture use taylor's theorem to prove $\sum_{k=0}^n \frac{1}{k!}<3$
woven radishBOT
SWR
dense venture
I know what taylor is but you are right, we are at the very beginning of calculus (the class?, not from an english speaking country), so we havn't had the taylor theorem yet
If you guys don't have an idea either I think I'm going to just show it numerical and hope that its fine like this
But thanks for your help @lusty sapphire and @dry geyser
lusty sapphire
That's my only guess if you can't use series or convergence or calculus
dense venture
Maybe a dumb question but what exactly is calculus, limits?
lusty sapphire
hmm... Do you know $\sum_{k=0}^n \frac{1}{2^k}<2$?
woven radishBOT
SWR
lusty sapphire
actually yeah. I think that will work
You can do it with just induction and inequalities
No convergence required
dry geyser
You can do it with just induction and inequalities
What about the first inequality?
lusty sapphire
Maybe a dumb question but what exactly is calculus, limits?
Calculus is the study of continuity. Specifically, regarding either instantaneous rates of change or area under a curve.
lusty sapphire
What about the first inequality?
That's still a thinker. But I have an approach for the second one, which is something at least
dry geyser
Yeah that’s true
dense venture
Calculus is the study of continuity. Specifically, regarding either instantaneou...
Oh okay so differentiation and integration, good to know
lusty sapphire
@dense venture @dry geyser okay I have a full solution now. You can solve the whole problem with just induction and inequalities (and very basic summation, which you need anyway since its part of the problem)
Honestly the second inequality was harder
dense venture
Whow nice
dry geyser
<@581520866911715328> <@678340429175062552> okay I have a full solution now. You...
I also just came up with an idea, haven’t fledged it out yet though
dense venture
I'm guessing a completely different approach to what I had started?
dry geyser
For me yes
lusty sapphire
I'm guessing a completely different approach to what I had started?
tbh I didn't read your attempt
dry geyser
LOL
dense venture
No problem 😄 there surely was a big mistake at some point anyways
dry geyser
Interesting
dense venture
So directly before the insertation from my side
dense venture
yeah the red arrow was wrong
But in theory it is possible to insert from the requirement or not?
dry geyser
I don’t think so no
< is transitive
So you can conclude a<c from a<b and b<c
But that wasn’t given here
You had a<b and a<c
And concluded that b<c
Which of course doesn’t work
dry geyser
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
My idea is to use n^(n-1) >= n! Btw
I think one can use that
But I’m not yet sure
dense venture
Wait, lets move my question with the insertation back
dense venture
My idea is to use n^(n-1) >= n! Btw
@lusty sapphire do you have the same attempt
lusty sapphire
nope
lusty sapphire
This is where we stayed in agreement
Looking at this, I actually diverged even earlier from you
@dense venture okay, which inequality do you want to work on first? The first or second one?
dense venture
The first (left) one was the one I started with but I don't have a preference
lusty sapphire
They'll both take about the same amount of effort, so we'll do the first one.
You've proven the base case. So now we will assume $\left(1+\frac1n\right)^n\le\sum_{k=0}^n\frac{1}{k!}$ is true for some specific, fixed $n$.
woven radishBOT
SWR
dense venture
Yes
lusty sapphire
With this fixed $n$ is mind, let us examine $\left(1+\frac{1}{n+1}\right)^{n+1}$
woven radishBOT
SWR
lusty sapphire
What do you think is the first thing we should do?
woven radishBOT
Marvin
lusty sapphire
I'd first seperate the n+1 case soooo $x^n * n$
don't add spaces inside the dollar sign opener and closer
Use $\cdot$ or $\times$ over $*$
woven radishBOT
SWR
lusty sapphire
Also I believe you meant $x^nx$
woven radishBOT
SWR
dense venture
Oh yeah should have spelled it out
lusty sapphire
Okay, with that separation, consider what we can do to make use of our $\left(1+\frac1n\right)^n\le\sum_{k=0}^n\frac{1}{k!}$ assumption.
woven radishBOT
SWR
dense venture
Probably inserting the right side of the inequality in the $(1+\frac{1}{n})^n$
woven radishBOT
Marvin
lusty sapphire
Probably inserting the right side of the inequality in the $(1+\frac{1}{n})^n$
That $\textit{would}$ work, but we don't have $\left(1+\frac{1}{n}\right)^n$ in our expression.
woven radishBOT
SWR
dense venture
Oh yeah the n+1 not n
lusty sapphire
But you're close. So give it some thought
dense venture
Can we do an index shift on the right side, and then insert the left side of the assumption
dense venture
But you're close. So give it some thought
Otherwise no idea how we could insert the right side into the left. Since there is the additional 1 in the denominator
lusty sapphire
Can we do an index shift on the right side, and then insert the left side of the...
index shift?
dense venture
$\sum_{k=0}^{n+1}\frac{1}{k!}$ -> $\sum_{k=0}^n\frac{1}{k!}+\frac{1}{(n+1)!} $
woven radishBOT
Marvin
lusty sapphire
$\sum_{k=0}^{n+1}\frac{1}{k!}$ -> $\sum_{k=0}^n\frac{1}{k!}+\frac{1}{(n+1)!}...
Not the approach I had, but this isn't a wrong statement. But I don't know if it will lead you to the solution yet. How would you continue from here?
dense venture
Played it out but that was at some point yesterday:
Insert the right side from the assumption $(1+\frac{1}{n})^n\le\sum_{k=0}^{n}\frac{1}{k}$ into the $\sum_{k=0}^n\frac{1}{k!}$
dry geyser
Played it out but that was at some point yesterday:
Insert the right side from ...
Put _{k=0}
woven radishBOT
Marvin
lusty sapphire
Played it out but that was at some point yesterday:
Insert the right side from ...
Did you forget a factorial?
That may work, but I predict it being messy. The fact you had trouble with it already seems to support that prediction
dense venture
Probably
dry geyser
Yeah I also think it leads to a dead end
dense venture
That _may_ work, but I predict it being messy. The fact you had trouble with it ...
What would have been your step
dry geyser
Im honestly very curious to see SWR‘s solution
lusty sapphire
We have this so far, right?
dense venture
yes
lusty sapphire
Try getting to this
They won't be equal. There will be inequalities, so think it through.
dense venture
So just the single 1 missing okay
lusty sapphire
So just the single 1 missing okay
"missing" could be a confusing way to think about it.
dense venture
Yeah, my first guess would be since the denominator is reduced the whole second pair of braces will be bigger so the whole term is increased
lusty sapphire
Yeah, my first guess would be since the denominator is reduced the whole second ...
yes. That is exactly right
dense venture
Perfect
lusty sapphire
So now you can use your inductive hypothesis.
Then you just need to do some algebra
dense venture
But thats only possible since we check if this whole term is smaller then something else
Since if the increased term is still smaller the not modified one would also be smaller
lusty sapphire
But thats only possible since we check if this whole term is smaller then someth...
what?
lusty sapphire
Since if the increased term is still smaller the not modified one would also be ...
what?
dry geyser
Then you just need to do some algebra
@lusty sapphire are you certain that this will yield the inequality?
dense venture
We have the original one with both n+1 in the denominator a
and the modified b
Since we check if a < c we can increase a to b
If b is smaller than c, a has to be smaller than c since a < b
lusty sapphire
<@374795910032523266> are you certain that this will yield the inequality?
I sent you a DM of my solution. You can verify if you like.
lusty sapphire
We have the original one with both n+1 in the denominator a
and the modified b
...
what are "a" and "the modified b"?
lusty sapphire
We have the original one with both n+1 in the denominator a
and the modified b
...
What we're going to be doing to show that a<b: We are first going to show that a<c. Then we will show that c<b. Then we can conclude that a<b.
@dense venture I have told you several lies.
@dry geyser has shown me some mistakes in my proof.
dense venture
I fear so too
dry geyser
It was the same you did (in a way) @dense venture
dense venture
Oh no thats why it looked so similar
dry geyser
If you can show that the sum from i to n divided by n+1 is smaller or equal to 1...
And the same I attempted here
I suppose maybe it makes sense to try my approach now, though I fear I got no clue where it leads
My idea is use the binomial theorem
dense venture
I guess we can try
dry geyser
And show that each element of the sum on the left is less than or equal to each element of the sum on the right
dense venture
So inserting the binomial definition into the $(1+\frac{1}{n+1})^n$ ?
woven radishBOT
Marvin
lusty sapphire
I'm gonna break to eat some food. Hopefully my head will be more clear after that
good luck to you two until then
dry geyser
$\frac{n!}{k!(n-k)!(n+1)^k}$
dense venture
Yes, bon appétit thanks for your help @lusty sapphire
We'll ping you when we get it or give up 😄
woven radishBOT
FirstNameLastName
woven radishBOT
FirstNameLastName
dense venture
Looks right
dry geyser
**FirstNameLastName**
I think the proof should be easy here
dense venture
But how do you get to that form in the first place 😺
dry geyser
But how do you get to that form in the first place 😺
You mean how do we prove the binomial theorem?
dense venture
You mean how do we prove the binomial theorem?
Thats not the problem but how to you get the initial inequality into this form guess its a bit to late for my brain
dry geyser
That’s not a form of the statement itself
It’s an element of the sum given by the binomial theorem
Replace the binomial coefficient with its definition
dry geyser
**FirstNameLastName**
And you get a sum involving this
So we don’t actually use induction
dense venture
So inserting my $(1+\frac{1}{n+1})^n$ into this
woven radishBOT
Marvin
dense venture
So not with n+1 just n
dry geyser
**FirstNameLastName**
I noticed it should probably not be n+1 in the denominator
dense venture
Okay yes I'll give it a shot
dry geyser
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
Basically, plug in the left side into the binomial theorem and you’ll get a sum where each element is smaller than or equal to each corresponding element of the sum in the middle
dense venture
**FirstNameLastName**
Likely a very obvious answer but where is the (n+1)^k coming from @dry geyser
dry geyser
Likely a very obvious answer but where is the (n+1)^k coming from <@678340429175...
Has to be n^k
Typo
dense venture
okay yeaj n^k makes sense
devout snowBOT
@dense venture Has your question been resolved?
dense venture
Germany
dense venture
Hätte die Kommunikation leichter gemacht 
dry geyser
Ich würde auch nicht mit < anfangen um ehrlich zu sein
Weil wir haben ja keine Induktion
lusty sapphire
Did we get the solution?
dry geyser
Ich würde durch umstellen (also einfach = ) zur erwünschten Form gelangen, und dann erst im letzten Schritt zeigen, dass die Ungleichung gilt
dense venture
Part a yes, just some problem in the fine print at the end
dry geyser
Did we get the solution?
I sent you the idea in dms
Only the „trivial“ part is missing
dry geyser
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
Was wir also zeigen wollen ist dass die linke Seite gleich der Summe ist die wir mit dem binomial theorem (keine Ahnung wies auf deutsch heißt) bestimmt haben
Danach kümmern wir uns um
Vorher brauchen wir kein Ungleichheitszeichen
dense venture
Also $=(1+\frac{1}{n})^n$ ?
woven radishBOT
Marvin
dense venture
Kann man nicht das theorem als gegeben annehmen und muss nicht noch die umformung beweisen
Oder habe ich es falsch verstanden
lusty sapphire
Did y'all do the second inequality yet?
dry geyser
Did y'all do the second inequality yet?
Not yet but I get the idea you had earlier
lusty sapphire
Then I'll let you Germans hash it out
dense venture
We can continue in english
dry geyser
Danach kümmern wir uns um
Du musst das Theorem nutzen um das hier zu bekommen
dense venture
Achso aus dem Linken das Recht machen
dry geyser
Nicht ganz, eine Sekunde
lusty sapphire
We can continue in english
For part 2, my hint is to consider $\sum_{k=2}^n\frac{1}{2^k}$
woven radishBOT
SWR
dense venture
Du musst das Theorem nutzen um das hier zu bekommen
Bin vermutlich ziemlich lost ich soll das theorem benutzen um von der linken Summe auf die rechte Summe zu kommen oder wie
lusty sapphire
Das ist die Idee
I understand this
idle swift
Heyy everyone, I have to prove that:
I tried to solve it by induction, the solu...
use k! > =2^k-1
u should have <1+ sumof (1+1/2+1/4+…)=3
dense venture
Das ist die Idee
Hat viel zu lange gedauert habe einen beweis der erstmal richtig scheint, gucke mir jetzt mal die zweite Ungleichung an. Ist mittlerweile was spät geworden
dry geyser
Hat viel zu lange gedauert habe einen beweis der erstmal richtig scheint, gucke ...
Verstehe
Können ja auch morgen fertig stellen wenn nötig
dry geyser
.close
devout snowBOT
dense venture
No need to reopen, we finished it @lusty sapphire your tip was very helpful but needed a few explainers by @dry geyser 😄
Also thanks for your tip @idle swift
devout snowBOT
Available help channel!
elder schooner
I need help with a geometrical and line/segment question
devout snowBOT
I need help with a geometrical and line/segment question
elder schooner
The line is straight, ABC is the line
B is the midpoint
AB = x^2 + 3x
BC = 2x+6
and i need help to solve that
elder schooner
that it represents the half of the line
so both sides are equal parts
x^2 + 3x = 2x+6
ik that
but i cant quite find out the x
quartz plaza
Yep, so now here you can try and solve for x
rare merlin
elder schooner
i tried the
uh
bocx
box and diamond method where i factor
but it didnt pan out
knotty sage
x^2 + 3x = 2x+6
how about subtracting 2x-6 from both sides then try and factor
knotty sage
Correct
elder schooner
then how do i continue
knotty sage
Suppose we have a quadratic with solutions a and b
(x-a)(x+b)=x^2+(-a+b)x-ab
elder schooner
i dont rlly understand that allat much
im in 10th grade geo
and quadratics i havent gotten to
knotty sage
i dont rlly understand that allat much
In simple terms
what two number multiplied together gives -6 and when added gives -1
knotty sage
It’ll look like this
(x-3)(x+2) and if you multiply it you’ll get the original quadratic we have
elder schooner
so its -3 then the x added on then the -2 and same?
ok so i got the (x-3)(x+2) then how do i place this into the equation
knotty sage
Substitute (x-3)(x+2) for the whole quadratic equation
elder schooner
sorry i have no idea how to do that
my bad for taking a min its a completely new idea given to me in my hw i was struggling the past hour to understand
knotty sage
So like we know
(x-3)(x+2)=x^2-x-6
then we can rewrite x^2-x-6 as (x-3)(x+2)
elder schooner
ohhh i see i see
knotty sage
in the original equation
elder schooner
so out of the -3 and 2 ik one is the answer
knotty sage
so out of the -3 and 2 ik one is the answer
not really
(x-3)(x+2)=0
the only way it’s zero is one of the expressions is zero or both
We can say either
x-3=0 or x+2=0
elder schooner
so x = -3 and x = 2
knotty sage
No
knotty sage
now one of these values when plugged in the original line equation it’ll give a negative value (ofc line cannot have a negative value)
and also lines cannot have lengths of 0
elder schooner
when i plug in the 3 into the original expression i get each side is 12 and AC = 24
but ik thats rlly off
however -2 gives me 10 and AC= 20
knotty sage
you calculated it wrong for both values
elder schooner
oh yea
(3)^2
9
- 3(3)
18
(3)2+6 = 12
18 and 12
but it cant be cus both sides have to be equal
knotty sage
Wait lemme check it rq
knotty sage
The mistake is here
3x-2x=x not -x
elder schooner
OHH i see
knotty sage
The process is still the same tho
knotty sage
yep so ab=-6 and a+b=1
elder schooner
x= 3 and x=2?
knotty sage
≠-6
knotty sage
x = -2 and x=3
yep
knotty sage
x-2 = 0 and x + 3 = 0
Yea this my bad
knotty sage
yep
elder schooner
(-3)^2 = 9 + (-3)3 = 18
and
(-3)2+6 still equals to 12
so 30
but is unequal sides
knotty sage
Multiplying by negative tho
elder schooner
?
knotty sage
-3(3)≠9
elder schooner
ohhh
yes yes mb
but those sides are unequal
so if i do it by 2
(2)^2 = 4 + 2(3) = 10
2(2) + 6
wait did i do smth wrong
wait no
knotty sage
(2)^2 = 4 + 2(3) = 10
it was x^2+3x
knotty sage
and for the other line
knotty sage
Yep
elder schooner
so its 2 and not -3
knotty sage
Then x=2
knotty sage
Yes
elder schooner
i see, thank you this helped a lot
elder schooner
lmaooo yeah well thank you for this
devout snowBOT
@elder schooner Has your question been resolved?
spark rivet
hi i am doing this problem but i am not sure where i went wrong
devout snowBOT
hi i am doing this problem but i am not sure where i went wrong
spark rivet
i put in 12.7 km/min and got it wrong
devout snowBOT
@spark rivet Has your question been resolved?
spark rivet
No
devout snowBOT
@spark rivet Has your question been resolved?
devout snowBOT
@spark rivet Has your question been resolved?
rustic zealot
Not sure what to do with 5. Answer for 4 was 40-32t-16h. I'm not sure what to do with just one set of data points :(
devout snowBOT
Not sure what to do with 5. Answer for 4 was 40-32t-16h. I'm not sure what to do...
finite helm
The heights at those times are 63.04 and 26.56. Divide the difference between those by the time difference and you have the average velocity - 30.4
devout snowBOT
@rustic zealot Has your question been resolved?
silk burrow
Hello, our math teacher gave us this extremely hard math problem, the worst part is that we have to prove it. I'm really lost.
A sheet of paper in the form of a rectangle ABCD with dimensions a × b, where a > b, is folded as in
so that point A merges with point C. Explain why, for the content S of the resulting pentagon PBCQR is 1
ab/2 < S < 3ab/4.
devout snowBOT
Hello, our math teacher gave us this extremely hard math problem, the worst part...
silk burrow
Tried to visualize it a bit
I think I have to combine the areas of BCR and CPQR to further prove it, but I know very little about the points PQR
Its not very clear in the picture, but CPQR and ARQB are the same shapes, just mirrored and shifted a little
full sand
shouldn't Q be on the line BC?
silk burrow
My bad, the second shape looks like this
silk burrow
now we know we only got out this part
So this part has to be bigger than ab/4 and smaller than ab/2 in every case
full sand
area(PRQC)=area(PRDA) btw
silk burrow
hint: the quadrilateral PRQC has area ab/2
Oh i see
that means PBC and CQR have the same area?
silk burrow
how come
silk burrow
area(PRQC)=area(PRDA) btw
if this is true?
full sand
ok look at the line PR; we must have R to the left of P
and PR passes through the midpoint
full sand
this means that area(ADR) is less than area(ADE) where E is midpoint of CD
similarly for CPB
and area(ADE) = ab/4
full sand
yeah
full sand
yeah same thing for CPB too
silk burrow
Why?
full sand
because RP is EF rotated slightly anticlockwise
You can reason in terms of angles too if you need
full sand
I mean, lengths and angles
silk burrow
Isnt it just that RP has to cross the midpoint of ABCD?
Oh I see it now
When I imagine it
full sand
APCR is a rhombus btw
silk burrow
R cant be on the right side of E since a>b
silk burrow
APCR is a rhombus btw
Good one
but we still dont have an exact lenght for the AP or RC line
but at least we know its bigger that 1/2 of abcd
wait we got it no?
full sand
Yeah we have it already
silk burrow
Since PBC=ADR=RCQ
full sand
yeah
silk burrow
And they are each smaller than 1/2
full sand
smaller than 1/4
silk burrow
(APE has to be larger than 1/4 of abcd because the rhombus is more than 1/2)
silk burrow
Therefore PCQR doesnt have area of 1/4
full sand
yeah rhombus has size at least 1/2 now that you mention it
full sand
np
silk burrow
Can we cover one more maybe when Im here?
We got 3 questions but I think I already got the third one
And while you are here
silk burrow
np
The natural numbers a, b are such that a > b, a + b is divisible by 9 and a-b is divisible by 11.
a) Determine the smallest possible value of a + b.
b) Prove that both a + 10b and b + 10a must be divisible by 99.
If you got time of course
silk burrow
I got my notes at school but I remember I got a) as 27
I got dinner, will be back in a moment sorry
full sand
np
devout snowBOT
@silk burrow Has your question been resolved?
silk burrow
express both a and b in terms of m and n
Am I correct?
full sand
one moment
full sand
write a+b=9m and a-b=11n
oops no add or subtract the two together to get 2a in terms of m and n, and 2b in terms of m and n
full sand
now add multiples of both together to find 2a + 20b in terms of m and n, same for 20a + 2b
then observe that 99 and 2 are relatively prime
then I think you've got b) done
silk burrow
what?
full sand
because it's 2(a+10b), 2(b+10a)
and we haven't shown that a=(9m+11n)/2 is a natural number
yet
or you can also assume that
find
a+10b=(?m+?n)/2
10a+b=(?m+?n)/2
silk burrow
oh we are going for be
okayy
a=4,5m+5,5n
b=4,5m-5,5n
Therefore
a+10b=49,5m - 49,5n
2a+20b=99m-99n
And 2b+20a= 99n+99m
??
full sand
yeah
you see how the 99 pops out
so 99 divides 2b+20a
then since 99 and 2 are relatively prime
99 divides b+10a as well
silk burrow
but wouldnt that be smtn.5?
full sand
we already know that b and a are natural numbers
so (b+10a) is also a natural number
and cannot be something.5
silk burrow
oh right
full sand
I don't remember how to do part a in a systematic way
full sand
lemme see
2a=9m+11n
2b=9m-11n
silk burrow
what?
full sand
just thinking
full sand
ok
silk burrow
no?
full sand
yeah I think you are right
silk burrow
now whats the mathematical solution for this
full sand
I want to say use something related to https://en.wikipedia.org/wiki/Bézout's_identity#Structure_of_solutions but I think just proof by exhaustion
full sand
yeah just trial and error
devout snowBOT
@silk burrow Has your question been resolved?
winged oracle
.reopen
azure plinth
hey quick question if we have a function h:R+→[1/4,+∞[ defined as h(x)=x+sqrtx+1/4 and we have its composition gof so we know f(x)=x+1/4. if we are asked to prove the surjection of f can we just the domain and co domain of h to prove that?
devout snowBOT
hey quick question if we have a function h:R+→[1/4,+∞[ defined as h(x)=x+sqrtx+1...
uncut crow
what do h and g have to do with the problem
and what is g
if we are asked to prove the surjection of f
and are there any typos here?
azure plinth
and what is g
they dont really add anything to the probleme i just wanted to know what domain and co domain we use for f when we have it as a composition of another function
uncut crow
azure plinth
> if we are asked to prove the surjection of f
and are there any typos here?
nope
so i suppose simply using the domain as R+ and codomain as [1/4,+∞[ should be fine?
devout snowBOT
@azure plinth Has your question been resolved?
uncut crow
i don't know what the question is
azure plinth
i don't know what the question is
okay lets say we got an h:E→F; x→x+2x+1 and we got f:x+2x and g:x+1 wich are the components of h as in h=fog so in this case what would the domain and codomain on f and g be ?
devout snowBOT
okay lets say we got an h:E→F; x→x+2x+1 and we got f:x+2x and g:x+1 wich are the...
azure plinth
idk if this makes sense or not i have been trying to find the answer for hours😭
uncut crow
devout snowBOT
@azure plinth Has your question been resolved?
sour spire
😭😭
sour spire
not really now
sour spire
Hi Dino you are extinct
i still have to study math
amber needle
i still have to study math
Are are undergrad
sour spire
Are are undergrad
not yet
amber needle
a country in Asia
Ok what you wanna do in next year
devout snowBOT
Available help channel!
delicate fossil
!close
devout snowBOT
sour spire
okaay
delicate fossil
😂
devout snowBOT
Available help channel!
formal quarry
can i have some help here?
devout snowBOT
can i have some help here?
formal quarry
there is no previous problem to look at
faint gorge
Change order of integration
formal quarry
Change order of integration
i tried to do that but i think i ended up with the wrong bounds
i had 2<x<0 and 0<y<x/2
restive river
Best to cite the source if you’ve taken it from somewhere else 😉
winged oracle
Best to cite the source if you’ve taken it from somewhere else 😉
?
runic prawn
this is ai generated isn't it
restive river
Yeah that’s why
runic prawn
altho the answer it gives doesn't seem bad
uncut crow
?
<@&268886789983436800> repeated chatgpt copy pastes and i told them it was against the rules already
restive river
I don’t discourage it (tho I think it’s banned from the server)
It’s best to atleast cite it such that we are all aware of possible inaccuracies
Ofc it’s a great tool to learn (I use it all the time)
autumn fjord
faint gorge
i tried to do that but i think i ended up with the wrong bounds
Did you draw it
formal quarry
2<x<0 doesnt make sense
should it be the other way around?
faint gorge
Yup
formal quarry
Hm
woven radishBOT
bacc (unhelpful)
faint gorge
Yup and then u-sub should deal with it
devout snowBOT
devout snowBOT
Available help channel!
red spire
hi, is |x| * |x| = x² = f(x) right?
and if I want to say that x² is differentiable at = 0, then I can just take f'(x) = 2x and insert x = 0, 2 * 0 = 0, so it is differentiable?
or is this an unprecise way to show that x² is differentiable?
runic prawn
what you said is correct but doesn't show x^2 is diff
to show that use the limit definition
red spire
I think I need to show it is differentiable for both sides
then it would be with lim and I think it's (h² - 0) / h = 0 for h -> 0
then I only showed it once hm
.close
devout snowBOT
devout snowBOT
Available help channel!
formal quarry
isnt this hella weird to ask me to graph this?
devout snowBOT
isnt this hella weird to ask me to graph this?
formal quarry
on a mditerm with no calculators and w/o any aid, just asking
acc ykw i can just plug in numbers and do it ig...
restive river
on a mditerm with no calculators and w/o any aid, just asking
Hmm not sure if that’s a question that someone could answer on this server 😅
formal quarry
im just asking cuz this is from a previous midterm and asking to graph this is a really weird graph
restive river
,w plot (x^2 -11x + 18)/(|x-9|)
woven radishBOT
acoustic leaf
not too bad if you factor the numerator first
formal quarry
true...
restive river
The modulus is really tickling my brain
Not sure how to factor that into the graph 😅
formal quarry
well a absolute value cant go in quadrants 3 and 4 so from there
acoustic leaf
you can use the piecewise definition of absolute value:
[ \abs x = \begin{cases*} x & if $x > 0$ \ 0 & if $x = 0$ \ -x & if $x < 0$ \end{cases*} ]
delicate fossil
take two cases i guess
fossil locust
oh isn't that just $\frac{(x - 9)(x - 2)}{|x - 9|} = \text{sgn}(x- 9) (x - 2)$
delicate fossil
x > 9, remove the modulus
woven radishBOT
cloud
fossil locust
yeah cloud told you
woven radishBOT
south's secret twin brother
devout snowBOT
@formal quarry Has your question been resolved?
pseudo ember
Wallah I'm here to solve
silent sandal
.close
devout snowBOT
devout snowBOT
Available help channel!
still rune
I know that this question involves using combinatorics
devout snowBOT
I know that this question involves using combinatorics
still rune
But I can't seem to figure out what I have to use
in terms of the formula to use, I just know that it has to be with no order for the first question so it wouldn't be with n^k or nPk
winter dome
There are 10 pets are are used in a)
So the possible arrangements of these pets are 10!
There will be 3 dogs and 2 chinchillas left
So we subtract those from the original amount leaving us with 5 dogs and 1 chinchillas
We have a total of 5 dogs, 1 chinchilla, 6 bunnies and 1 cat. We add them together to get 5 +1 +6 +1= 13 animals
Which is an excess
So we do 13! -3!
Is the answer correct?
devout snowBOT
@still rune Has your question been resolved?
plush grail
devout snowBOT
plush grail
because we're differentiating with respect to y
stone fjord
yes
devout snowBOT
devout snowBOT
Available help channel!
sturdy zenith
in changing variables for integration
devout snowBOT
in changing variables for integration
torn bane
and you use the absolute value of the jacobian right?
the absolute value of the determinant
sturdy zenith
is determinant not the same as jacobian
but thank you
I want to clarify just in case
torn bane
is determinant not the same as jacobian
the jacobian refers to the matrix formed by the partials of the transformation
devout snowBOT
@sturdy zenith Has your question been resolved?
acoustic leaf
either the matrix or its determinant can be called the jacobian, which is a bit confusing
devout snowBOT
devout snowBOT
Available help channel!
soft parrot
A certain number has 60 factors. Amongst these are the numbers 6, 7, 8, 9 and 10.
Find the number.
devout snowBOT
A certain number has 60 factors. Amongst these are the numbers 6, 7, 8, 9 and 10...
soft parrot
pls help
sand dove
the number of factors can be easily found with prime factorization
knotty sage
We can conclude that
1,2,3,4,5 are factors too
sand dove
the number of factors can be easily found with prime factorization
for example, if A = 2^3 * 3^5, you have (3+1)(5+1) = 24 total positive factors
now your mystery number can be written as prime factorization too
and you already know some of the primes (if not all primes) that will intervene in that factorisation
as well as their minimum valuation
(in our example A = 2^3 * 3^5, 2 has valuation 3)
sand dove
pls help
so let's run through this first fact:
what are the primes we definitely know this number is divisible by?
soft parrot
i’m confused
sand dove
i’m confused
is there anything you're particularly confused about?
soft parrot
wai
dry geyser
what are the primes we definitely know this number is divisible by?
This is the same as asking what is the smallest number divisible by all given factors
dry geyser
what to do after prime factorisation
Send what you’ve got
sand dove
nvm idk how to start
you know your mystery number, call it A, is divisible by 6 through 10
soft parrot
oh yeh
sand dove
what is the biggest number we can make with those that A is divisible by
soft parrot
100
sand dove
uh
sand dove
did you do 6*...*10?
dry geyser
what is the biggest number we can make with those that A is divisible by
You mean smallest?
sand dove
You mean smallest?
no I did mean biggest
dry geyser
Wait no that doesn’t make sense either
sand dove
lcm
soft parrot
did you do 6*...\*10?
6x7x8x9x10
sand dove
6x7x8x9x10
just because A is divisible by all of those doesn't mean A is divisible by their product
12 is divisible by 4
12 is divisible by 6
is 12 divisible by 4*6?
sand dove
so when A is divisible by 6,7,...,10, you can't assume A is divisible by their product
soft parrot
oHHh
sand dove
but A is indeed divisible by a number that is a common multiple to all of those
since A is a common multiple of those numbers itself
it's divisible by...
sand dove
yes
sand dove
yes
and it's useful and easy to compute the lcm with prime factorization
for example when you want the lcm of say, 2^3, 2*3^2 and 3*5
for each prime you always keep the highest exponent (called valuation) you find
so for 2, the highest exponent yields 2^3
for 3, it's 3^2,
for 5, it's 5^1
sand dove
for example when you want the lcm of say, 2^3, 2\*3^2 and 3*5
and so the lcm of that list is 2^3 * 3^2 * 5
sand dove
alright
sand dove
for example, if A = 2^3 * 3^5, you have (3+1)(5+1) = 24 total positive factors
perhaps you can use this formula to find the number of (positive) factors of 2520?
(feel free to question this formula if you don't know how/why it's true)
sand dove
no :(
soft parrot
wah 😭😭
sand dove
for example, if A = 2^3 * 3^5, you have (3+1)(5+1) = 24 total positive factors
did you understand how 24 was gotten with this example?
soft parrot
uh
prime factorise 2520
then
count the factors
WIST
i did it wrong
i think
2520
2 1760
2 880
2 440
2 220
2 110
2 55
5 11
11 1
ong no
i did it wrong
dry geyser
Do you know what the prime factorization of 2520 is?
sand dove
in the same way we could have reached 2520, or 2^3 * 3^2 * 5 * 7
I mean I gave you the prime factorization, but if we have to go through why this is true