Also I got various solutions with various methods
#help-13
paper karma
opal hinge
,rccw
wraith daggerBOT
crimson sedge
Yes?
paper karma
guide me solve
spring forum
!15min
cedar kilnBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
paper karma
its already been a long
paper karma
please be patient
oh ohk!
spring forum
I don't really have time rn, but my intuition for 49 is just to plug the values in
and good luck on your hw!
paper karma
I don't really have time rn, but my intuition for 49 is just to plug the values ...
i mean to solve it
spring forum
yes, plug the values in for the first step
paper karma
what if option wouldnt have been given man
spring forum
then use the cos(theta)=sin(pi/2-theta)
use this identity to make cos 15x into sin([some angle]). what do you get
@paper karma
paper karma
yep
spring forum
what do you get?
paper karma
pi/2-15x
spring forum
yes, now what's the whole equation?
paper karma
sin(pi/2-15x)=sin5x
spring forum
right, now subtract sin5x from both sides and use the sum to product identities
and what do you get?
spring forum
mhm
paper karma
so its
so eq^n is sin(pi/4-5x)(sin(pi/4-10x))
ty now i can do it myself further ty so much
spring forum
so eq^n is sin(pi/4-5x)(sin(pi/4-10x))
ty now i can do it myself further ty so m...
wait wait reminder
paper karma
pls
wraith daggerBOT
αστραία 💫 (astraea)
spring forum
it's not sin*sin, it's 2*cos*sin
paper karma
oh sry sillies my god
spring forum
no problem! 
paper karma
thank you, for ur precious time n help
paper karma
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
smoky zephyr
.
(x+y) * z = a
(x*y) + z = a
give me five numbers which satify
one such example is
x=2,y = 1,z = 1
x = 4,y =3,z = 2
smoky zephyr
how to prove it mathematically
the pattern i observed was x = 2z and x = y+1
to find any such numbers
(n,n-1) n/2
indigo lagoon
$(x+y) \times z = a$
$(x\times y) + z = a$
give me five numbers which satisfies
one such example is
x=2,y = 1,z = 1
x = 4,y =3,z = 2
surreal cave
well if x=0 and y=1 then you have infinitely many possible z's but I presume you want non-zero positive integers?
indigo lagoon
nvm, it's not better
hot cipher
(x+y) * z = a
(x*y) + z = a
give me five numbers which satify
one such example ...
Btw, you can find a general formula for it cause they are set equal to the same number
I love transitivity
smoky zephyr
Btw, you can find a general formula for it cause they are set equal to the same ...
did not get it
hot cipher
if $ (x+y)z = a$ and $xy+z=a$, then $(x+y)z = xy+z$
wraith daggerBOT
zzz0nnn
smoky zephyr
if $ (x+y)z = a$ and $xy+z=a$, then $(x+y)z = xy+z$
then
hot cipher
pull out a z = (some combination of x and y)
You could do it with any variable, but thats what i did anyways
smoky zephyr
pull out a z = (some combination of x and y)
elabrote please
it will become xz + yz = xy + z
put z = .......
smoky zephyr
Try to separate z from x and y from there, its pretty easy
we substitute xy in the above equations?
i dont think it works
hot cipher
You know distribution, so you know how to take a factor out
Can you see any common factor from the left side?
crimson sedge
xz + yz - z = xy
z(x+y) - z
hot cipher
z(x+y) - z
Yes, also, but you still havent isolated the z
crimson sedge
Yes, also, but you still havent isolated the z
z(x+y) - z = xy
z - z = xy/x+y?
hot cipher
it should be z - z / x+y = xy/x+y there
crimson sedge
I ate up a step but that's the same
crimson sedge
it should be z - z / x+y = xy/x+y there
If u r taking it this way then it should be z(x+y) - z/(x+y) = xy/x+y
hot cipher
not either
crimson sedge
So where is it wrong and why explain it
hot cipher
$\frac{z(x+y)-z}{x+y}=\frac{xy}{x+y}\implies z-\frac{z}{x+y}=\frac{xy}{x+y}$
wraith daggerBOT
zzz0nnn
hot cipher
z(x+y) / x+y cancel to z
You set z / x+y = z, which is clearly not true
unless x+y = 1 
crimson sedge
Exception exists lol
hot cipher
again, the easiest way:
$z(x+y) - z = xy \implies z(x+y-1) = xy \implies z = \frac{xy}{x+y-1}$
wraith daggerBOT
zzz0nnn
crimson sedge
Ohh yeah you are correct
hot cipher
Choosing x+y =/= 1, you get any value you want that satisfies the original system of equations
crimson sedge
True
crimson sedge
again, the easiest way:
$z(x+y) - z = xy \implies z(x+y-1) = xy \implies z = \f...
But how do you know this much like writing it in a code way
You must have a great experience on these things
hot cipher
Kind of, its called LaTeX, its a math writing language
-> https://latexeditor.lagrida.com/ if for some reason you wanna test with it
pulsar lily
technically it's a typesetting language
I learned it first in this server. there are several examples of latex things people have written in this server, and overleaf has great documentation on how to actually use it
if you are a math student, you should learn it
indigo lagoon
This channel is turning into tex help as well
@smoky zephyr Are you still here with us?
cedar kilnBOT
@smoky zephyr Has your question been resolved?
flat mica
**zzz0nnn**
this is not transitiivty btw
copper owl
Hi, I just want to check if i did it right
f: A -> B, A = {1,2,...,m}, B={1,2,...,n}
I need to find how many functions f are monotonous increasing
I asked for help earlier with a similar problem and I got adviced to try to change it to something I understand, so I made this:
Being B = {amount of adopted dogs}
and A = {days}
Then we have n dogs and m days.
The day m, n dogs get adopted
The day m-1, n-1 dogs get adopted
and so on until day 0 where 0 dogs get adopted
so the answer would be n! * m!
is that alright?
chrome elk
Do we want the functions to be strictly increasing or nah
copper owl
no need to
pulsar lily
then there's no need for f(m) = n
copper owl
why
pulsar lily
if it need not be strictly increasing, then f(a) = 1 for all a in A is valid
chrome elk
We can represent functions from [m] to anything as sequences
And we want it to be a weakly increasing sequence
copper owl
maybe i got it wrong
i wanted to say that it is f(a_1)<=f(a_2) (a_1<a_2)
copper owl
yes
chrome elk
Well I'm still thinking
Oh yeah we can turn this into another sequence using the indices
chrome elk
There is absolutely no way the answer is symmetrical
chrome elk
A lot of combi stuff is just seeing things and using them later
I don't see how one would come up with a lot of stuff on their own
chrome elk
Oh yeah we can turn this into another sequence using the indices
Anyway continuing off this
b_i = a_i + (i-1)
Gives another sequence
1 ≤ b_1 ≤ ... ≤ b_m ≤ n + m - 1
These two are in bijection I'm pretty sure
So now it's just about picking m elements from n+m-1
chrome elk
I barely understand it myself, I'm using all my combi experience here
Which isn't a lot
But essentially
A function [m] to [n] which is monotonic increasing
Is essentially an increasing sequence of m length from [n]
copper owl
okay
chrome elk
So we write it the way I dod
But that doesn't give us anything helpful directly cuz the entries can be equal
So we create another sequence, this time using the indices
And this sequence is strictly increasing (prove this)
copper owl
okay
ill try to but i have to leave for a few minuts
ill come back if needed
thank u
have a great day
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
chrome elk
. @copper owl close the new channel you claimed and come back here
copper owl
here i am
wicked mantle
chrome elk
What parts of it
copper owl
And this sequence is strictly increasing (prove this)
how to follow after this
chrome elk
You have a strictly increasing sequence in [n+m-1]
The sequence has m elements
So you're done, it's just a choice now
copper owl
why is the sequence n+m-1
solid ridge
Hi im new
chrome elk
Hi
solid ridge
I didnt know this server existed
solid ridge
This place looks awesome
cedar kilnBOT
This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.
solid ridge
Oh mb
copper owl
What can you tell about b_m now
b_m < n
chrome elk
...no
copper owl
🙁
chrome elk
b_m is a_m + m - 1
chrome elk
Then you have a strictly increasing sequence in [n+m-1]
And this corresponds to monotonic increasing functions [m] to [n]
chrome elk
Then you have a strictly increasing sequence in [n+m-1]
And this is very easy to count
chrome elk
You need to prove a few things though
You need to prove that the b's form an increasing sequence
And that the mapping from a_i to b_i is a bijection of sequences
But I'm gonna leave that to you
cedar kilnBOT
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Available help channel!
loud moth
i received this puzzle from my math teacher and i've been stuck on it for about a week. every solution i try ends up in a quartic equation (and we're not allowed to use the quartic formula to solve it), could anyone help?
loud moth
i dont understand how i could try solving it without ending up in a quartic equation
h also has to be at its maximum (since the line could also be more horizontal)
tropic oxide
oh this looks mega ugly
i do think that you will in fact run up against the quartic wall here, no matter what
last apex
even if you get a quartic you can solve it without the quartic formula
loud moth
i do think that you will in fact run up against the quartic wall here, no matter...
he says he's managed to solve it without running into a quartic
tropic oxide
say, what is the quartic you're getting
last apex
especially if the numbers are carefully chosen
loud moth
even if you get a quartic you can solve it without the quartic formula
we havent been taught how to solve quartics
let me get my notebook real quick just a moment
ok so for my work
i divided h into 1 and a
h being 1+a
eventually, i end up at:
a^4 + 2a^3 - 7a^2 + 2a + 1 = 0
tropic oxide
ok so in fact this is a "nice" quartic
divide both sides by a^2 and you might notice sth
last apex
look at (a + 1/a)^2
loud moth
a^2 + 2 + 1/a^2
last apex
the two is in the -7!
loud moth
so i end up with (a + 1/a)^2 - 9 + 2a + 2/a = 0
tropic oxide
what we're building up to is the idea to substitute x := a + 1/a
loud moth
(a + 1/a)^2 - 2 ( 4.5 + a + 1/a) = 0
?
then x = a + 1/a
x^2 - 2(4.5+x) = 0
x^2 - 2x - 9 = 0
tropic oxide
you didnt rly need to tamper with the nine at all and also uh
loud moth
ah
tropic oxide
x^2 + 2x - 9 = 0
loud moth
what'd i miss why the plus
tropic oxide
-9 + 2a + 2/a ≠ -2(4.5 + a + 1/a)
loud moth
i end up with x = -1 - sqrt(10) v x= -1 + sqrt(10)
and the maximum is -1 + sqrt(10)
so the height is 1 + ( -1 + sqrt(10))
which is just sqrt 10
last apex
remember that x = a + 1/a
last apex
so before you add the 1 solve for a
loud moth
a + 1/a = -1 + sqrt(10)
do i multiply by a here
a^2 + a = -a + a*sqrt(10)
a^2 + 2a - a*sqrt(10) = 0
OHH
a ( a + 2 - sqrt(10)) = 0
a = 0 v a + 2 - sqrt(10) = o
a cant be 0, so a = sqrt(10) - 2
height is therefore
sqrt(10) -2 + 1
sqrt (10) - 1
wait huh that cant be right
he said the height was around 2.5
this is
uh
~2.16
livid hound
1/a * a isn't a
loud moth
OH
whoops
a^2 + 1 = -a + asqrt(10)
a^2 +a -asqrt(10) + 1 = 0
ok im stuck again
how do i solve this one
loud moth
what do i do with the a - asqrt(10)
ohh nevermind
i got it
thank you guys!!!
!solved
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clever glacier
For which natural numbers n does the inequality 2^n > n^3 hold? Prove your statement.
Do I solve normally with proof by induction? Can’t seem to find a general method to solve/prove this.
ancient lodge
For which natural numbers n does the inequality 2^n > n^3 hold? Prove your state...
yeah induction would work
the idea is that 2^n grows faster than n^3 for sufficiently large n
so you can guess when this will hold true by playing around
and then prove it by induction
native heath
cedar kilnBOT
@clever glacier Has your question been resolved?
cedar kilnBOT
@clever glacier Has your question been resolved?
flat mica
For which natural numbers n does the inequality 2^n > n^3 hold? Prove your state...
you can try using known logarithm inequalities
$$\log x \le \frac{x^s}{s} \text{ for } s > 0$$
$$\log x \ge 1 - \frac 1 x$$
wraith daggerBOT
gfauxpas
flat mica
namely, the second gives you log 2 >= 1/2
clever glacier
namely, the second gives you log 2 >= 1/2
Oh wow, wait why log? To estimate where n is true or?
clever glacier
and then prove it by induction
Tysm!!
flat mica
Oh wow, wait why log? To estimate where n is true or?
because you can take log of both sides of an inequality (where terms are >0)
and reverse it by taking e^ both sides
meaning it's sufficient to prove the inequality that you get when taking log of both sides
clever glacier
meaning it's sufficient to prove the inequality that you get when taking log of ...
Ok I see! I’ll try it in my proof…tomorrow😁 tysmmm
cedar kilnBOT
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Available help channel!
flint wave
Hello, I would like assistance on understanding a linear relations question, having to do with solving it algebraically. Here is a photo of the question for reference
flint wave
Question 5
night wedge
You can avoid doing the same question over and over with different numbers by generalising to any slope m you want.
Then for each m you use it to find 2 points Q
But you have to do undefined differently
flint wave
Okay, I know the formula y2-y1/x2-x1, I’m just confused on how to use it in this circumstance
This whole unit is brand new to me
night wedge
$m=\frac{y_{2}-y_{1}}{x_{1}-x_{2}}$
wraith daggerBOT
BBMaths
night wedge
Oops I messed it up
flint wave
Shouldn’t the x’s be flipped
night wedge
X1 and x2 wrong way
flint wave
Yea
night wedge
You can let x_1 and y_1 be -1 and 2
flint wave
So do I substitute -1,2 for the y1 and x1 values
night wedge
Yeah
flint wave
Got it, let me write that in
night wedge
Then for x_2 you pick what you want as long as it isn’t -1 (not x_1 again)
rain idol
Wait could you just use
$$ y_1 - y = m(x_1 - x) $$
wraith daggerBOT
lily
flint wave
Okay, for sign rules I came out with y2-2/x2+1
rain idol
It's the point-slope formula
night wedge
That’s a really useful formula that does what you want
rain idol
also I accidently flipped the subscript
night wedge
Doesn’t matter it just *-1 both sides so it still works
flint wave
So it helps better for when I’m trying to find alternate slope values?
rain idol
So it helps better for when I’m trying to find alternate slope values?
Yes
You can solve for it by plugging in the values you do have
flint wave
Okay
rain idol
Example in Desmos
flint wave
Wait this can be done on a graphing calculator?
rain idol
Yes
night wedge
Yes
flint wave
I’m just wondering if my teacher wants me to learn how to do it algebraically though
night wedge
The equations are algebra
flint wave
I meant with the original formula
rain idol
$$
y - y_1 = m(x - x_1) \
$$
$$
y - 2 = m(x - (-1)) \
$$
y - 2 = mx + 1 \
$$
$$
y = mx + 3 \
$$
Find $m$
i'm so rusty in latex
flint wave
What happens when the number becomes a fraction
wraith daggerBOT
lily
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
rain idol
What happens when the number becomes a fraction
Simplify if possible, if not leave as is
flint wave
For example 1/3 for my slope
flint wave
And it can work with the same formula
rain idol
Yes, just use the distributive property
flint wave
My graphing calculator doesn’t have an y button, only x,0, and T
So when I try to put it in as a graph it shows me y1=
rain idol
Honestly you really don't need a graphing calculator for this, these are linear equations
flint wave
Okay, so I’ll just stick to paper
So what I have right now is, my setup, I’m going to put a random x value and then solve for y
Do you want me to replace m with 2
rain idol
Since you already have a point (-1, 2), x1 = -1 and y1 = 2. You only need to plug in m and the resulting points should be your answer
The equation takes a point and a slope to create a linear function
flint wave
I tried to roughly simplify it, I put x2 as 4, I got y=12
After I subbed m for 2
I don’t know if I’m understanding the question though
So my final answer for one out of the two examples was (4,12)
flint wave
I think they want me to just try random values for x2 and then solve for y2
rain idol
"Slope of PQ" is an implict product
flint wave
Because I checked (4,12) and the key said it was correct
rain idol
I see
flint wave
0,4 is correct aswell
So I understand that part now
But my next part of the question wants me to do it with a fraction, 1/3
rain idol
Ah so it's a point-slope problem
I mean if you plug it into the point-slope formula then you're already done
flint wave
I’m just confused how I’m going to deal with a fraction on the other side
Haven’t dealt with that much
I’m going to but x2 as 4 again and see what happens
rain idol
(0, 7/3) would be the one for the 1/3 slope
flint wave
I got y2= 7/3 for x2=4
rain idol
I got this after solving
night wedge
Just so it’s clear the numbers after letters don’t mean anything and if you write it down it should be smaller and in bottom right like $x_{2}$
wraith daggerBOT
BBMaths
rain idol
Just so it’s clear the numbers after letters don’t mean anything and if you writ...
For typing them on Discord an underscore between the variable and number is sufficent
ex. x_1
night wedge
Not on paper though
flint wave
I’m not suppose to do it on paper?
rain idol
Write it as so $x_1$
wraith daggerBOT
lily
night wedge
maybe I’m being nit picky but I would make sure it’s clear so you don’t double it by accident ¯_(ツ)_/¯
rain idol
Yes
night wedge
I got this after solving
Back to the question now you just use this ^ and put in any x you want except x=-1
flint wave
Because I can just take (-1,2) and move it up 1 and right 3
Getting my two values out of that
wraith daggerBOT
lily
rain idol
As so
flint wave
Kinda hard to visualize without a graph😅
Okay I got (2,3) and (5,4)
Key said that was correct
Now with 0
I’m guessing I just do the same thing but m is 0
rain idol
Zero is a horizontal line
rain idol
It's just (0, 2) since the slope is null
flint wave
But there can be more than one value, right
Because I need two
Oh wait I get it, y will ALWAYS be 2
rain idol
Well it's a flat line you could just pick a random point in the domain
flint wave
so I can just pick a random x value
rain idol
Yes
rain idol
If the rise is unknown multiply the run and the slope. If the run is unknown then use the slope to find it.
For (a)
$$ \frac{rise}{49} = \frac{5}{7} $$
$$ {rise} = \frac{5}{7} \cdot 49 = 35 $$
For (d)
$$ -\frac{3}{4} = \frac{4}{run} $$
$$ -3 \cdot {run} = 4 \cdot 4 $$
$$ -3 \cdot {run} = 16 $$
$$ {run} = -\frac{16}{3} $$
wraith daggerBOT
lily
flint wave
ooh okay, just times them by eachother to find the missing value
rain idol
Exactly, just solve
flint wave
for c in question 6, I times them and I got a fraction, what do I need to fix
rain idol
Wait what step are you on of solving it
Oh wait nvm
I know what you mean
that's fine the answer can be a fraction
since it's just asking you for the third measure (e.g. rise or run)
rain idol
I would simplify, but yes
rain idol
Looks good to me
flint wave
they answer key says -18
rain idol
weird?
flint wave
im confused
rain idol
You must've miscalculated
night wedge
.solved (try this)
night wedge
I guess you just own this channel now
flint wave
.solved
cedar kilnBOT
night wedge
Oh I see
flint wave
there we go
rain idol
Ok well good luck
flint wave
Thank yo have a great rest of your day/night
cedar kilnBOT
Available help channel!
night wedge
Maybe not entirely related to mathematics but perhaps in a meta way, does anyone know all the commands like !occupied etc. for use in the help section
dull oxide
Maybe not entirely related to mathematics but perhaps in a meta way, does anyone...
status
da2a
occupied
xy or original
show
help
night wedge
.solved
cedar kilnBOT
cedar kilnBOT
.solved
If you are done with this channel, please mark your problem as solved by typing .close
dull oxide
I forgot that one
cedar kilnBOT
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ocean rover
cedar kilnBOT
ocean rover
can I please ask for you to help check where I went wrong in my working? For bottom left my sign for d1n and d2n seem to be flipped from the answer sheet. I did the long matrix multiplication way, but I checked everything and it looks okay.
cedar kilnBOT
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abstract pollen
cedar kilnBOT
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spice phoenix
cedar kilnBOT
spice phoenix
Wtf? That's not the pic
fossil dawn
resend the actual pic
spice phoenix
spice phoenix
It was glitching out for me idk why
spice phoenix
Yeah showed a random pic for me
fossil dawn
ah
spice phoenix
I showed this to my professor, and she said I did this wrong, because I pulled a t out of a 4, which didn't have a t
fossil dawn
wait a sec
spice phoenix
Yeah it is
spice phoenix
I meant t mb
fossil dawn
oh
final river
$\lim_{t \to \infty}$ ?
spice phoenix
Yes
wraith daggerBOT
holathere
fossil dawn
what did you do here
esp in the numerator
i see you divided the numerator by t, but also multiplied it by t
spice phoenix
I rewrote the 4, to take out a t, and cancel it with the numerator and denominator
final river
All that did was just move a denominator term up to the top
spice phoenix
So I didnt need to rewrite 4, I just move the t from the denominator to the top?
final river
I don't think all of that would've mattered anyways because as t gets large that 4/t^2 will go to 0 which doesn't really affect the stuff in the parentheses
And then the t outside the parentheses in the denominator becomes infinitely large bringing the product with it
Meaning the entire expression goes to...
cedar kilnBOT
@spice phoenix Has your question been resolved?
sharp violet
For calc 3, im struggling to go through finding vector equations for planes. I understand how to get it for lines, but I am not getting how to do it for planes.
lofty sand
For calc 3, im struggling to go through finding vector equations for planes. I u...
Let’s do an example
sharp violet
Find an equation of the plane.
the plane that passes through the point
(2, 3, 2)
and contains the line of intersection of the planes
x + 2y + 3z = 1 and 2x − y + z = −3
So i know that our r0 is going to be 2,3,2, but I am not sure where we can get our normal plane from.
lofty sand
A plane is perfectly described by :
1 point
2 independent vectors
Right ?
You have a point, now it’s time to find your vectors
Can you find the line where the 2 planes intersect ?
sharp violet
Can we add the two vectors together to find the intersection?
No dumb question
We can take the vectors and cross product them to get a vector.
That in turn is out normal vector no?
lofty sand
Are you talking about finding the intersection of the 2 planes ?
sharp violet
Im rapidly trying to how to get the normal vector of each plane, because if we cross product the normal vectors, we can get the intersection
lofty sand
The normal vectors are not hard to find they’re the coefficients
sharp violet
Is that only when the planes themselves are perpendicular to the plane passing through?
For a line it makes sense, if the line is perpendicular to the plane, the equation of the plane would be perpendicular, and becoems the normal vector
lofty sand
I think it’s better to just solve
Say z=k is your parameter
And then just solve for x & y
sharp violet
Im not sure I understand, are you saying plug in our first point into each equation?
cedar kilnBOT
@sharp violet Has your question been resolved?
fallen coyote
cedar kilnBOT
fallen coyote
can you help walk me through finding t and r?
loud gyro
you know how the vectors add and subtract right?
fallen coyote
somewhat, but I can't remember fully
loud gyro
can you tell what you know?
fallen coyote
oh wait yeah, you just add and subtract individual components
fallen coyote
is it you connect the tips of two vectors?
loud gyro
yep
fallen coyote
its going from u to v, so would i do v-u
loud gyro
yes
loud gyro
yep
fallen coyote
ohhh
so i can do that when it connects the tips of two vectors, right?
how about r?
loud gyro
do you know parallelogram law?
fallen coyote
hmm i dont think so
loud gyro
fallen coyote
ohhh
loud gyro
you can divide you problem into steps, since its in 3d
fallen coyote
so i can just add the numbers regularly again?
loud gyro
and do the parallelogram
loud gyro
so i can just add the numbers regularly again?
yep
fallen coyote
you can divide you problem into steps, since its in 3d
oh, could you show me how?
loud gyro
r vector is the diagonal of the cube formed with u, v, w right?
fallen coyote
yeah
loud gyro
you can try making a parallelogram that includes r as a diagonal, and one for the sides as one of the vectors u,v,w
Hint:||the vector s would form the other side of the parallelogram if you take v as a side|| ||u = s+v||
fallen coyote
Hint:||the vector s would form the other side of the parallelogram if you take v...
how so? they dont make a parallelogram
fallen coyote
hmm maybe I'll get it if I see it written out?
loud gyro
try to mark them in the diagram itself
visual representation is always better
like I already said, the two vectors would give you three vertices, and r being the sum would get you the 4th vertex of the parallelogram
just draw the sides and youd see
neon harbor
Could anyone help me with this math problem I'm like so lost I don't know the formula for it 😭
tropic oxide
do you know how to find the volume of a prism in general?
neon harbor
Yes
tropic oxide
ok, tell me how that's done
neon harbor
area of the triangle x height
neon harbor
Uh is it 9 x 12 x 1/2 ?
knotty fern
hello guys any tips for remembering multiple formulas ?
neon harbor
so in fact you do know the formula(s)
What do they have in common tho
livid hound
wdym
neon harbor
Uh Im still not sure about how to find the volume 😭
livid hound
you have the area of the triangle (the cross section)
multiply that by the height of the prism (or based on this orientation depth)
tropic oxide
hello guys any tips for remembering multiple formulas ?
!occupied
cedar kilnBOT
hello guys any tips for remembering multiple formulas ?
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).
neon harbor
multiply that by the height of the prism (or based on this orientation depth)
Ohhh I see I see thankss
.close
cedar kilnBOT
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sick cipher
cedar kilnBOT
cedar kilnBOT
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crude skiff
he
crude skiff
find the equations of direct common tangetns of the following pair of circles x^2 +y^2 -4x -2y + 1 = 0 and x^2 + y^2 +6x + 5 = 0
cedar kilnBOT
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.
fresh sage
Oh
My fault, haven’t helped for a while
Ok, so if u notice
Both the circles have a equal radius
night wedge
That’s not clear from the equations
fresh sage
Their direct common tangents are two parallel lines one on each “side” of the pair
crude skiff
Both the circles have a equal radius
Yes
crude skiff
Their direct common tangents are two parallel lines one on each “side” of the pa...
IK that, but how do i calculate and find it now?
i have the centres of both circle
and their radius
pastel vault
i have the centres of both circle
then the gradient will be the same as that of the segment connecting their centres
crude skiff
then the gradient will be the same as that of the segment connecting their centr...
yup, which comes out to be 1/5
but what do i do next!?!? I have got no other point to write the eqns of tangent
fresh sage
Yeah Im at 1/5 right now
crude skiff
Yeah Im at 1/5 right now
Mhmm
crude skiff
Yes
fresh sage
Ok
crude skiff
the distance comes out to be root26
pastel vault
so you could consider the perpendicular line going through the centre of one of the circles
for example, y - 1 = -5(x - 2)
and then the two intersections with the circle lie on the respective lines
crude skiff
I got 2 different possibilities
yeah?
pastel vault
y = \frac{1}{5} x - 1.44
should be surds
crude skiff
i got the answer is surd forms, should i tell you that?
fresh sage
Oh
pastel vault
fresh sage
Mines was the two direct common tangents
fresh sage
yours seem correct too ig
Yeah but I think ur way/ @pastel vault Is correct
night wedge
I had a question like this on the MAT and I got it wrong even after getting the right answer halfway through so I’m getting nightmarish flashbacks looking at this question
pastel vault
yeah I feel that yes it's a lot of algebra
night wedge
There are 4 lines that are tangent to both circles, but maybe the direct ones are the ones above
pastel vault
but conceptually this makes the most sense
crude skiff
https://www.desmos.com/calculator/6mepbfrio0
yeahh nice thing
crude skiff
There are 4 lines that are tangent to both circles, but maybe the direct ones ar...
two TCT, and two DCT
oh parallel, nvm
night wedge
I didn’t learn that terminology
crude skiff
I didn’t learn that terminology
I was talking about direct common tangent and transverse common tangent
crude skiff
Yeah but I think ur way/ <@1151031274413228043> Is correct
yeah
lemme try the perpendicular line thingy
night wedge
oh parallel, nvm
Sorry I meant tangent
crude skiff
OMG , my brain if fogged right now, i will do this after a quick cycle ride 🫠
pastel vault
actually you know, the indirect common tangent is easier
first find the midpoint of the centres
then sub y = m * (x + 1/2) + 1/2 into x^2 + y^2 + 6x + 5 = 0, then you need the discriminant to equal 0
you get a quadratic in m
this is why when the radii are the same, you have these nice conditions
it'd be a bit hellish otherwise
cedar kilnBOT
@crude skiff Has your question been resolved?
cedar kilnBOT
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karmic field
would proof to this be easy in any way or no
cedar kilnBOT
would proof to this be easy in any way or no
ancient lodge
eh it's just symbol pushing so yeah
ancient lodge
it's all fairly mechanical calculation with algebra and differentiation
there's no shocking insight or anything like that
karmic field
there's no shocking insight or anything like that
well it does seem so
because by looking at it
im not seeing the proof in my head
imma give it a bit more time to see if i can figure it out
btw the e^....
is a factor right ?
like we have multiplication there
ancient lodge
like we have multiplication there
yes you are multiplying by $\exp \left(-\int^{t}_{t_0} p(t) \dd{t} \right)$
wraith daggerBOT
Civil Service Pigeon
karmic field
p(x) dx
anyways small issue idk with notation
when we use W(y1,y2)(to)
we get a matrix that has y1'(to)
hollow trail
a proof from my textbook (spoilers if you want to prove it yourself)
karmic field
a proof from my textbook (spoilers if you want to prove it yourself)
ill check it after i finisht trying to prove it myself
hollow trail
not (y1(to))'
yes
hollow trail
wdym?
karmic field
because if i differentiate
its =>
since going backwards adds c
which isnt valid
so it only goes => the right way
so if i start with this and get to something thats true by differentiating it wont be a proof
as i cant go <=
i brought it to -ln (...) = $\int^{t}_{t_0} p(x) \dd{x} \right)$
wraith daggerBOT
Taebek
Compile Error! Click the reaction for more information.
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karmic field
but i realised i cant differentiate
hollow trail
it might help to note that you have already seen functions involving some e^(integral p(x) dx), probably at the beginning of the course
it might help to recall what context you saw that in before
karmic field
it might help to note that you have already seen functions involving some e^(int...
in first order linear ode solution?
hollow trail
yes exactly
karmic field
yea but that wasnt the integral function
hollow trail
so that sort of indicates that W(x,y) is the solution to some first order ode involving p
karmic field
so that sort of indicates that W(x,y) is the solution to some first order ode in...
idk
well i know y1, y2 , y1+y2 are solutions
i have an idea
asumme there is a solution y3(t) = u(t) y1(t)?
nvm this doesnt reduce the original one
and another problem is its p(x) not p(t)
hollow trail
x is just a dummy variable inside the integral
karmic field
that i cant get rid of
cause i cant differentiate it
should i maybe go through the solving progress
because this is just Dx/D
to solve for c1/c2
hollow trail
so we have this idea that maybe W is the solution to some first order linear ode. maybe if we take a look at what equation W' should satisfy we could recover that ode (and then we would be able to solve it for W)
karmic field
so we have this idea that maybe W is the solution to some first order linear ode...
W'(y1,y2)(t) = 0?
or did i make a mistake
hollow trail
no, it's more complicated than that
hollow trail
it isn't true
try writing out W(y1,y2)(t) in full and then taking the derivative of that
karmic field
Where did i make a mistake
I expanded it and replaced y1" and 2 with what they are equal to based on the 2nd order
karmic field
try writing out W(y1,y2)(t) in full and then taking the derivative of that
I did but i ended up on 0 for some reason
cedar kilnBOT
@karmic field Has your question been resolved?
karmic field
how are you crossing out these parts?
oh yea thats the mistake
p(t) y + y' = 0
is the equation of which W(y1,y2)(t) is a solution
hollow trail
yes
hollow trail
what is the issue?
hollow trail
the main thing it's useful for is to see that W(y1,y2)(t) is either:
- zero everywhere if c = 0
- nonzero everywhere if c =/= 0
it can also be used in the calculations for variation of parameters
cedar kilnBOT
@karmic field Has your question been resolved?
cedar kilnBOT
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Available help channel!
halcyon wyvern
What am I doing wrong with this excel formula? My prof wants me to use a relative frequency formula that can apply to the rest of the variables, but I just can't get it down.
hollow trail
if the value is in "" then it will search for matches to that string, so currently it is counting the number of cells containing the string "H18"
probably what you want is just the cell number H18, which would then count the number of cells matching the contents of cell H18
halcyon wyvern
h18 is the number 10, so it should count the amount of 10s in the data set right?
but whenever I press enter it gives me a 0. there are 3 tens in the data set
and whenever I drag the formula down I get 0s
im so bad at math trying to figure out how to use excel on top of that is torture
hollow trail
=countif(B:B,H18) counts the number of cells whose contents match the contents of cell H18. =countif(B:B,"H18") counts the number of cells whose contents are the string "H18"
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wild plaza
im stuck at the part where when u/(43+u^2) becomes (1/2)ln(43+u^2). how do i do that
stiff brook
substitution of k = 43 + u^2
wild plaza
oh i substitute it again?
stiff brook
oh i substitute it again?
yes
stiff brook
i think its just to remove the 2
no
stiff brook
the integral has a u in the numerator
rotund sapphire
I'm talking about the 1/2
stiff brook
you want to cancel it out and get a 1/k dk form
rotund sapphire
I know why the ln is there
stiff brook
the OP was asking about how they got that result. it is through a substitution.
stiff brook
the 1/2 also comes from the substitution.
wild plaza
.close
cedar kilnBOT
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lucid maple
how do i solve this kind of sistem
a = max((6+b)/ 2, 5)
b = max((a+c)/ 2, 3)
c = max((b+d)/ 2, 2)
d = max((c+1)/ 2, 4)``` where the numbers on the right are some given constants and the 6 and 1 from a and d are also given constantsand i need to do this for n variables
tropic oxide
!xy
cedar kilnBOT
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.
lucid maple
so x1 = max((constant + x2) / 2, constant)
xi = max((x(i-1) + x(i+1)) / 2, constant)
xn = max((x(n-1) + constant) / 2, constant)
its not from a textbook or from somewhere exactly
cedar kilnBOT
@lucid maple Has your question been resolved?
tropic oxide
its not from a textbook or from somewhere exactly
so you made the problem up yourself?
lucid maple
yeah i reduced another thing to this
what i had before is irrelevant
i just need a + b + c + d
crimson delta
if you ignore the max condition, then this is a linear system with n equations and n variables
I cant tell at a glance whether the coeff matrix is invertible
but feels like it
lucid maple
if you ignore the max condition, then this is a linear system with n equations a...
yeah theres def. a solution for the linear sistem
crimson delta
you could formulate it as a LP
and then use a LP solver
i.e. you have the conditions a>=(6+b)/2, a>=5, b>=(a+c)/2, b>=3 etc
lucid maple
whats an lp?
latent gull
cedar kilnBOT
latent gull
I want to know what will the integral be
??
I did calculate it and its wrong 
<@&286206848099549185>
rotund sapphire
<@&286206848099549185>
ping when 15 minutes have passed
latent gull
u need to add a constant term
Can you send me the solution
cedar kilnBOT
Can you send me the solution
The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.
cedar kilnBOT
supple coral
Alright <:catthumbsup:614540188747563008> still thanks for the help 🙂
You do realise they could have still helped you
latent gull
You do realise they could have still helped you
Oh no i ve got it know well idid a silly mistake and its alright
🫡
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Available help channel!
sullen spoke
How does this get simplified into 2x + 6
cedar kilnBOT
How does this get simplified into 2x + 6
cedar kilnBOT
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.
sullen spoke
I do not know if I am being gaslit or something
Did I miss this during school?? Is the curriculum in my country just bad that I didn't learn this??
loud gyro
,w factorize 2x^2+2x-12
wraith daggerBOT
floral arrow
That's wrong
loud gyro
you see, the factorization they did is wrong
floral arrow
Also what does that even mean
sullen spoke
Then what even is the correct answer??
loud gyro
the explanation looks like AI tbh
sullen spoke
Also what does that even mean
They tried factoring the denominator... which just returns the denominator
idk why
prisma pike
Did I miss this during school?? Is the curriculum in my country just bad that I ...
Yea it's a country issue ig
Rip for you man
sullen spoke
Yea it's a country issue ig
This isn't even an online quiz, it's a reviewer(unofficial) for a college admission exam
loud gyro
Their answer is correct tho
upon factorization you can divide the numerator and denominator by (x-2) and you get (2x+6) as the answer
floral arrow
$\frac{2x^2+2x-12}{x-2} = \frac{2(x+3)(x-2)}{x-2} = 2(x+3) = 2x+6$
wraith daggerBOT
Nel
floral arrow
That's what they tried to do
frail citrus
sullen spoke
upon factorization you can divide the numerator and denominator by (x-2) and you...
Could you show me the process for that?
sullen spoke
(x-3)(x+4) is the wrong numerator for this equation??
floral arrow
That's wrong
Like I said
tardy zenith
Help pls
supple hemlock
you doing ncert?
lavish vine
Help pls
Cannot find any errors apart from the fact that it might be circular reasoning
rotund sapphire
the criteria provided is wrong
It will not be AAS
Because im pretty sure AAS is applicable when top ∠ is used, and bottom ∠ with base is used
and here that is not the case
Other than that I can't really find defects
slender atlas
I think AAS is correct here, because a side that's not in between the two given angles is given.
hot cipher
AAS is satisfied sadly
slender atlas
I'm more inclined to go with circular reasoning.
rotund sapphire
hmm
hot cipher
The non-included side is the one they physically share (AD)
cedar kilnBOT
@tardy zenith Has your question been resolved?
hot cipher
Help pls
Yeah, i didnt wanna believe
But the defect is that we simply assumed that
ADperpBC if and only if ADC=ADB
tardy zenith
Brother what is answer
hot cipher
the third element of the reasoning
Basically: You didnt use the hypothesis properly
ADC = ADB is a thesis
At this, the angleA = angleC falls under that problem too
tardy zenith
I not understnd
hot cipher
I not understnd
Its poorly written, basically.
The ideas behind it are all sound, its just that we assumed a few things on the way without actually stating it.
cedar kilnBOT
@tardy zenith Has your question been resolved?
lavish vine
I not understnd
We claim that the angles opposite to equal sides are equal
It would have been better to use RHS congruency
nocturne spear
is this integration by parts?
cedar kilnBOT
is this integration by parts?
sacred iron
nocturne spear
is the integral solveable using substitution
long pawn
idt so
nocturne spear
because of the ln right
long pawn
besides by part is the easiest way here since the ln(u) makes for a great u function in udv - int(vdu)
long pawn
because of the ln right
yup!
long pawn
because of the ln right
the acronym that i would use in deciding which would be the u function in general is LIPET - logarithmic, inverse trig, polynomial, exponential, trigonometric, but that's just how I was taught it
past canopy
isnt it ILATE
nocturne spear
the acronym that i would use in deciding which would be the u function in genera...
how does that help you
long pawn
isnt it ILATE
i think there's multiple, that's why i said that's how i was taught it
long pawn
how does that help you
in an integration by parts, you'll have two functions multiplied - you have to pick one to be u and one to be v
if you don't pick the easier one to be u, life can get difficult
safe cloud
True or false question, I put false but the answer was true?
safe cloud
just confused bc like unless given that b=0, how can it be an even function?
green crater
have you tried to just calculate the integrals?
green crater
ye there is a cleverer option
but calculating would make u sure that this is true
for the cleverer option
safe cloud
but how could I see that without spending ages on it
green crater
first do u know abt function parity?
safe cloud
not by name
tropic oxide
odd and even functions
green crater
ye mb