And just to be clear, my problem here is that I don't know where to start to solve it
#help-13
inland ether
lemme think of a approach
id prefer to do it by value inputs but that's too time taking
chrome quail
This is just a reduced form of am-gm
inland ether
Can i use the proof by contradiction
try
inland ether
This is just a reduced form of am-gm
reduced?
carmine cloud
This is just a reduced form of am-gm
We didn't student Am Gm yet tho so idk what is it
unreal pumice
Maybe induction?
chrome quail
General form is that $\frac{n_1+n_2+...}{n} \geq (n_1 * n_2 *...)^(1/n)$
wraith daggerBOT
Stitches
inland ether
ik man
chrome quail
I mean, it might be begging the premise, but try proving $\frac{a+b}{2} \geq \sqrt{ab}$. You can then move to the n = 4 case trivially.
inland ether
this is its form at n=4 so i dont consider it reduced or smth else but depends on povs yk
wraith daggerBOT
Stitches
chrome quail
Hint for the two-case would be to start by squaring $\sqrt{a} - \sqrt{b}$. This isn't an induction question since we only prove it for 4 elements (induction would be apt for n elements)
wraith daggerBOT
Stitches
carmine cloud
I tried to do the proof by contradiction it didn't lead to anything,
carmine cloud
Hint for the two-case would be to start by squaring $\sqrt{a} - \sqrt{b}$. This ...
And then move to the abcd?
chrome quail
Don't try to contradict, start with $(\sqrt{a} - \sqrt{b})^2 \geq 0$
carmine cloud
Ohhh
chrome quail
Expand that and you get a direct proof
that proves it for the ab case, abcd follows fairly trivially
carmine cloud
Yeah yeah it makes sense
wraith daggerBOT
Stitches
carmine cloud
I did it, Now, should I just replace a with (a+b) and b with (c+d)?
chrome quail
(a+b)/2 to be precise but yeah
carmine cloud
Hey
edgy spade
Do you know the AM GM inequality?
carmine cloud
Nope, we didn't study it yet
edgy spade
Well okay do you know what jensens inequality is
carmine cloud
No?
This exercice was given to us after studying limits and continuity
(we did also in the lessons nth root and arctan)
Oh I got it I got it
Thanks
.close
small elbow
cedar kilnBOT
small elbow
what is the best way of approaching this prbolem?
crimson delta
just do the product?
well the obvious way would be to just go "well if you switch the coordinates via tau and then via sigma then you switch them as sigma circ tau"
in other words, instead of as matrices, think about it as linear maps
plush dock
let a,b,c \in natural numbers, such that \gcd(a,b,c)=1
prove that \gcd(a^3+abc,b^3+abc,c^3+abc)<=2
how do i completely prove this?
i have tried to prove it by contradiction, by assuming there is a prime p such that it divides a^3+abc,b^3+abc,c^3+abc and i can find that there can be atmost one of a,b,c that is divisible by p, so i broke it down into 2 cases:
- the case where (WLOG) a is divisible by p, one can prove that this easily results in a contradiction.
- the case where none of a,b,c is divisible by p
the case 2 is where i am stuck on right now, i am unable to find a contradiction here.
cedar kilnBOT
let a,b,c \in natural numbers, such that \gcd(a,b,c)=1
prove that \gcd(a^3+abc,...
@plush dock Has your question been resolved?
cedar kilnBOT
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tired vine
Get the support reaction of the beam
cedar kilnBOT
Get the support reaction of the beam
tired vine
I have the answer i just need help confirming it
cedar kilnBOT
@tired vine Has your question been resolved?
cedar kilnBOT
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oak anchor
How do you determine the determinant of a 3x3 matrix?
cedar kilnBOT
How do you determine the determinant of a 3x3 matrix?
last apex
There are formulas (which are the definition)
So this an easy one because this matrix is singular
So det =0
oak anchor
hm, so I have to remember this formula? There is no other way?
I thought you could find some way using the inverse matrix
crimson delta
finding the inverse matrix is way harder
oak anchor
finding the inverse matrix is way harder
I find it rather easy
crimson delta
if you write that sum in a different way you get sarrus rule
which is easier to remember
oak anchor
I see, thanks
crimson delta
how is finding the inverse by solving a system easier than computing a sum with 6 terms
digital cliff
its easier to just remember the stuff in the brackets are the minors of the matrix using the first row rather than in that form
crimson delta
more generally, the formula in that picture is obtained from using the laplace expansion on the first row. you can also use that expansion on a different row/column which is sometimes easier depending on where 0s are placed
grizzled vale
you can also obtain the determinant via row reduction but cofactor expansion is better for small** matrices that don't row reduce nicely
crimson delta
you can also obtain the determinant via row reduction but cofactor expansion is ...
for 3x3 it doesnt matter much but for any bigger matrix, as long as it doesnt have lots of zeros, you should never use the cofactor expansion instead of row reducing
blazing haven
hm, so I have to remember this formula? There is no other way?
You can just remember it by crossing out the top row, and then a column going left to right, then you just take the determinant of that two by two times whatever number is intersected on the top row by the two lines, then just shift your cross out column one to the right
tropic oxide
why you all so aggressive
how was that aggressive
grizzled vale
for 3x3 it doesnt matter much but for any bigger matrix, as long as it doesnt ha...
i mean ideally you use matlab or something for bigger matrices anyway but yeah, cofactor gets annoying quickly
blazing haven
Also determinants are inherently better, as they take less flops for a computer to compute than inverses
formal glacier
how was that aggressive
oak anchor
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cedar kilnBOT
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prisma pollen
how can an equilateral triangle be divided into 6 isoceles triangles
prisma pollen
it can be divided into 3 triangles if we extend a line from its centroid to its vertices
formal glacier
how can an equilateral triangle be divided into 6 isoceles triangles
draw the incircle for it
the points of contact divide it into 3 isoceles triangles
as tangents from an external point to a circle are equal
prisma pollen
the points of contact divide it into 3 isoceles triangles
how are these isoceles
formal glacier
how are these isoceles
mark the points of contact
prisma pollen
how are these isoceles
AEF?
prisma pollen
what 😭
prisma pollen
whats pure isoceles and just isoceles
its a triangle in which 2 sides are equal thats it
formal glacier
its a triangle in which 2 sides are equal thats it
do yk what a circumcircle is
find the circumcentre for the triangle
and join it with all the three vertices
that should make you 3 triangles all isoceles
prisma pollen
find the circumcentre for the triangle
the circumcenter of an equilateral triangle is the same as its incenter so the center in my diagram is also the circumcenter
formal glacier
the circumcenter of an equilateral triangle is the same as its incenter so the c...
so join em
whats stopping you
EAB , EDB and EDA
prisma pollen
EAB , EDB and EDA
we want to divide the figure into six isoceles triangles not 3
we have already divided it into 3 isoceles triangles
formal glacier
it can be divided into 3 triangles if we extend a line from its centroid to its ...
??
formal glacier
how can an equilateral triangle be divided into 6 isoceles triangles
ohh
I might need some coffee
lemme think
formal glacier
centroid = circumcenter = incenter in an equilateral tiraangel
hmm maybe we can use bpt here
simmilarity i mean
ancient nova
how can an equilateral triangle be divided into 6 isoceles triangles
I don't want to interrupt just want to ask do all 6 triangles have to be congruent
prisma pollen
<:emojigg_Huh:1398783215405367409> I don't want to interrupt just want to ask do...
no
formal glacier
its a combi probelm
huh
formal glacier
it can be divided into 3 triangles if we extend a line from its centroid to its ...
mb my brain wasn't working
here as i said before draw an incircle
and join the side's point of contact with the incentre
then just join the point of contacts
formal glacier
its a combi probelm
what's the actual question then ??
cedar kilnBOT
@prisma pollen Has your question been resolved?
rapid island
cedar kilnBOT
glass oasis
show us what you have done so far
glass oasis
still show us
rapid island
i cant find my phone
i'll just explain
i took a cuboid with its length 'a', breadth 'a', and height 'b', where its TSA minus the area of the top would be c^2
which didnt lead me anywhere
the top
rapid island
still show us
see 😞
ancient nova
Can you simplify the equation
nova glacier
You just have one square of area a^2 plus 4 rectangles of area ab. Together they need to be c^2.
And you want to maximize the volume, which is a^2 b
rapid island
You just have one square of area a^2 plus 4 rectangles of area ab. Together they...
woah yeah wait
but still
what i did should've given me the same wouldnt it
nova glacier
Simplify, find maximum
you had a mistake there
you need to subtract the top
which is a^2 not 2(b+a)
nova glacier
yw
cedar kilnBOT
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Available help channel!
gritty galleon
Let $f(x,y) = x^2 + y^2 + 2gx + 2fy + c = 0$ be the equation of a circle. If $f(0, \lambda) =0$ has equal roots $\lambda = 2,2$ and $f(\lambda,0)$ has roots $\lambda = 0.8,5$, then the centre of the circle is?
wraith daggerBOT
rak³en
gritty galleon
Progress till now:
found the ordinate of the centre to be 2 using some mathematical reasoning
so then using formulae its easy to observe f=-2
tropic oxide
the circle is tangent to the y-axis at (0,2) and intersects the x-axis at (0.8, 0) and (5, 0)
gritty galleon
the circle is tangent to the y-axis at (0,2) and intersects the x-axis at (0.8, ...
....done that alr lol
tropic oxide
or are you artificially restricted from using that kind of reasoning
gritty galleon
found the ordinate of the centre to be 2 using some mathematical reasoning
thats literally how i got this ...
now y-intercept = 2 = 2 sqrt(f^2-c)
so c=3
which means 5-0.8 = 2sqrt(g^2-c) (x intercept)
which means g^2 = 7.41
where instead i should be getting 8.41
tropic oxide
your mistake was proceeding with this absolutely fugly solution method
gritty galleon
your mistake was proceeding with this absolutely fugly solution method
honestly i would just put 0.8 and 5 as x into the equation and be done with it
but i think this is the 'better' (as in faster) method
gritty galleon
where instead i should be getting 8.41
know this because i've done this before by using the three points and just solving a system of equations
and my teacher confirmed it was right
tropic oxide
i mean why not do it geometrically
for convenience call the two known pts on the x-axis A and B and that on the y-axis C
we know the center lies on the perp bisector of AB, which is x = (0.8+5)/2 = 2.9
gritty galleon
oh right midpoint of AB is x of centre
tropic oxide
and we know the center lies on the line perp to the y-axis going thru c
gritty galleon
we know the center lies on the perp bisector of AB, which is x = (0.8+5)/2 = 2.9
yeah that just hit me like a brick wall
gritty galleon
and we get the coords of the center _instantly_
good point
that is probably the faster method i should use
thanks
gritty galleon
now y-intercept = 2 = 2 sqrt(f^2-c)
in any case tho i'd love to know whats wrong with this
i'll look into later ig
thx
.close
vital knoll
Why we use epsilon as example to explain that R is totally order field?
vital knoll
In my perspective
Even if we define the smallest positive number, it can be compared to some value a
Im wrong?
And from that i also conclude that there no positive number less than epsilon only 0 if we define that 0 is positive?
dire geode
what exactly is your argument
Even if we define the smallest positive number, it can be compared to some value a
what are you trying to do with this sentence
vital knoll
> Even if we define the smallest positive number, it can be compared to some val...
Prove that R is totally ordered field
Mean every value in R can be compared
vital knoll
Why we use epsilon as example to explain that R is totally order field?
And the book i read put this example
dire geode
explain what
vital knoll
Why we use epsilon as example to explain that R is totally order field?
Why that example ?
Why epsilon
dire geode
why what example
there's 2 of them and neither of them are proving R is totally ordered
vital knoll
My question is why epsilon?
dire geode
that's not a coherent question
vital knoll
What is the meaning of this example?
dire geode
this is saying that if such an a exists, then a must be 0. without knowing R is totally ordered, you can't conclude a=0
vital knoll
this is saying that if such an a exists, then a must be 0. without knowing R is ...
Say it from the beginning then
dire geode
wut
vital knoll
Then you understand my question
cedar kilnBOT
dire geode
lmao
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Available help channel!
wary heath
Does this proof look right? I've always found it hard to keep track of my variables so I want to make sure I'm saying this correctly. Thank you.
void glen
looks good to me
misty locust
I don't understand the way to solve it
cedar kilnBOT
I don't understand the way to solve it
misty locust
I have 24k and 24l but it doesn't quite work when i try to put it as k*l=18 or i am doing it wrong way
languid charm
Wdym by it doesnt quite work?
misty locust
Wdym by it doesnt quite work?
Because i multiplied it all and got 10368
And i think there's less common multiple but i need a proper way to get it
Whatever i will ask teacher about it
.close
cedar kilnBOT
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Available help channel!
livid hound
Who understand this derivative and can explain?
cedar kilnBOT
Who understand this derivative and can explain?
tropic oxide
whoever wrote this made a rather serious notation error
tropic oxide
they didnt actually put the derivative symbol, somehow
and so we would have been led to believe that the function somehow equals its own derivative (which it doesn't)
anyway, looking at only the first step, can you point out exactly what's confusing you
tropic oxide
$\frac{\sqrt{3x}}{2}$ is convenient to split up as $\frac{\sqrt{3}}{2} \cdot \sqrt{x}$ in order to take its derivative (as you'd have to do as part of the chain rule --- the 2nd factor is the derivative of the inside function)
wraith daggerBOT
Ann
tropic oxide
$\frac{\sqrt{3}}{2}$ is a constant after all, and $\sqrt{x}$ is something you either know or can look up the derivative of
livid hound
$\frac{\sqrt{3x}}{2}$ is convenient to split up as $\frac{\sqrt{3}}{2} \cdot \sq...
oh but its also a compound function?
wraith daggerBOT
Ann
tropic oxide
oh but its also a compound function?
what exactly are you talking about when you say "it"
livid hound
what _exactly_ are you talking about when you say "it"
root 3x/2
tropic oxide
gonna need you to be careful with brackets here.
unless you want me to tell you exactly how ambiguous root 3x/2 is.
livid hound
gonna need you to be careful with brackets here.
so everything thats not here is compoud function?
tropic oxide
anyway, if you WANT to view sqrt(3x)/2 as a composition of two other functions, nothing is stopping you from doing so.
tropic oxide
so everything thats not here is compoud function?
no, rather this table is written in such a way that discards the notion of "composite functions" in general and prefers to make the table twice as big while listing every single tabular derivative both on its own and with an inner function.
in your case, the one you're interested in is arcsin[u(x)] and the u(x) is sqrt(3x)/2.
livid hound
in your case, the one you're interested in is `arcsin[u(x)]` and the u(x) is `sq...
but why does he split the root in two?
tropic oxide
to make it more convenient to differentiate.
hybrid hornet
so that he can differentiate it more easily
tropic oxide
there are like 20 different ways to make your own life very difficult and painful and full of suffering
breaking up sqrt(3x)/2 as sqrt(3)/2 * sqrt(x) is a way to not do that
livid hound
breaking up sqrt(3x)/2 as sqrt(3)/2 * sqrt(x) is a way to **not do that**
i dont understand anything
tropic oxide
there are many different ways to calculate the derivative of sqrt(3x)/2
some ways are easy and some ways are stupid hard and will make you cry
hybrid hornet
breaking up sqrt(3x)/2 as sqrt(3)/2 * sqrt(x) is a way to **not do that**
seems pretty ok tbh
tropic oxide
do you understand me now
livid hound
more or less
tropic oxide
seems pretty ok tbh
yeah because as i said thats a way to not have great pain and suffering
tropic oxide
more or less
ok wonderful.
livid hound
but try to explain less complicate my english is not very good
tropic oxide
breaking sqrt(3x)/2 into sqrt(3)/2 * sqrt(x) is how you AVOID trouble and pain
hybrid hornet
you mean why sqrt(x) turns into 1/(2*sqrt(x)) ?
livid hound
you mean why sqrt(x) turns into 1/(2*sqrt(x)) ?
nono the step down
livid hound
yes
tropic oxide
and do you mean why, or how?
livid hound
why and how
livid hound
on down its not by 2
tropic oxide
@hybrid hornet would you mind taking this over fully
stuff is happening on my end irl
livid hound
what he does is this
but if u do by 2 its not a 4*3x
livid hound
OHHH
but one question if i forget the *2/2 what hapens
like if i dont simplify it in the exam
livid hound
oh and in the last step
he separates 1/2 bc its not root
and root up/root down = big root?
livid hound
no im asking if my explanation is correct
hybrid hornet
yes its correct
hybrid hornet
2nd row is actually a solution, the rest is just making look better
just making solution cleaner
if you can, do it
hybrid hornet
if you have the solution its good
sometimes my teachers are ok when I keep the solution
just the long ugly one
livid hound
i will ask her tmrw
livid hound
do u hapen to know when to use logarigmic derivation?
?
i think my complications are with problems
i cant show u bc there in my language
hybrid hornet
not really I only use (lnx)' = 1/x
livid hound
when is it used?
hybrid hornet
I can do one example
livid hound
when u have like x+5 to the power of 2x?
hybrid hornet
when u have like x+5 to the power of 2x?
yes like that
hybrid hornet
no problem
hybrid hornet
just show
wind stone
catalan???
livid hound
if i have like find tangent y=3x^3+2 line paralel to the line y=3x+2 for ex how to do
livid hound
catalan???
how did u know?
wind stone
how did u know?
i speak it
livid hound
i speak it
where u from?
upper ruin
Omg, I didn't know it was that big of a difference from Spanish 😲
wind stone
Omg, I didn't know it was that big of a difference from Spanish 😲
its similar but imo harder
upper ruin
By reading it from the photo, it would seem a mixture of Spanish and Portuguese
wind stone
i thought so too until i realized i could read and understand it LMAOOO
hybrid hornet
chain rule
livid hound
the sqrt(1-x^2)
livid hound
and then derive f(x)
hybrid hornet
then multiply f(x)
livid hound
and multiply
hybrid hornet
yes
livid hound
ohh
hybrid hornet
in this case you have 2^(arcsin(x))
livid hound
but thats the chain rule or only this formula?
hybrid hornet
so you have 2^(arcsin(x)) times ln(2) * derivative of arcsinx
hybrid hornet
but thats the chain rule or only this formula?
its by chain rule
livid hound
chain rule is f[g(x)]*f'(x)?
livid hound
not really
livid hound
this yes
hybrid hornet
and you want derivative of that
this is chain rule
so 2^(arcsin(x))
you first take derivative from "outside" times derivative of "inside"
livid hound
like explanation i dont understand but i can 50/50 apply it
hybrid hornet
derivative of 2^(anything) is equal to 2^(anything)*ln(2)
and now derivative of inside which is arcsinx
derivative of arcsinx is just the 1/sqrt(1-x²)
hybrid hornet
yes
so if you have
derivative of cos( sin(x) )
what is derivative of cos? -sin
so its -sin(sin(x))
livid hound
for ex (2x+3)^4 = 4(2x+3)^3 * 2?
hybrid hornet
exactly
upper ruin
Yep
hybrid hornet
yes
livid hound
so i derive it like that
hybrid hornet
yes
livid hound
yea
hybrid hornet
alright
nvm
so in the 2nd part where he derives
sqrt(1-x²)
so like how I said
hybrid hornet
no problem hahahaha
livid hound
tysm man
hybrid hornet
you're welcome
hybrid hornet
nice nice
hybrid hornet
yeah I just had that 2 weeks ago
anyways gtg if you have no questions type .close
livid hound
tysm man
hybrid hornet
yw
livid hound
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
twilit tapir
cedar kilnBOT
pulsar lily
it's probably because they knew that $8^{1/3}=\sqrt[3]{8}=2$
wraith daggerBOT
Flip
tropic oxide
$a^{m/n} = (a^{1/n})^m$ is fairly standard
wraith daggerBOT
Ann
pulsar lily
does that look familiar?
twilit tapir
ohhhh
yes
I got this
I think I messed up there
it’s meant to be bracketed?
the square of 2 goes outside the. Bracket
tropic oxide
,rccw
wraith daggerBOT
tropic oxide
you didn't mess up but your 2/3 exponent is written in a bad way
tho that's bad handwriting not bad math
twilit tapir
okay okay thanks alot also
I’m confuse about how I’d go about
Doing
64 to the power of -0.5
pulsar lily
no, what you have should still be equivalent. you instead wrote $8^{2/3}=(8^2)^{1/3}=64^{1/3}=\sqrt[3]{64}$
pulsar lily
where?
twilit tapir
Would I do root 64
twilit tapir
64 to the power of -0.5
.
pulsar lily
it seems like you're confusing a cube root for a square root
twilit tapir
wraith daggerBOT
Flip
pulsar lily
oh, sure
pulsar lily
looks good to me
pulsar lily
go on
pulsar lily
do you have a collection of exponent rules?
pulsar lily
what looks relevant to you?
pulsar lily
can you explain it in more detail?
pulsar lily
if you start with just (1/2)^4, you should be able to only use exponent properties to obtain a number
pulsar lily
which property are you invoking here?
twilit tapir
I’m soooo lost
pulsar lily
notice how in this example, the solution writers manipulate the given expression using only properties of exponents to obtain a number: 4
twilit tapir
oh yes
upper ruin
I’m soooo lost
What would 2⁴ be?
upper ruin
Awesome
twilit tapir
I LOVE U ALBERTO
pulsar lily
what we want from you is the same approach, using properties of exponents to arrive at a number
twilit tapir
that's the idea
okay how do we do so
twilit tapir
Awesome
If it was (1/3)^4
would I do 3 to the power of 4
and then get the reciprocal
pulsar lily
and then get the reciprocal
it sounds like you're picturing this as like, a disconnected sequence of operations
but we're not really "doing" anything. we're just applying rules to a number to get a chain of statements
(1/5)^12 = (5^(-1))^12 = 5^(-12) = (5^12)^(-1) = 1/(5^12)
that's one way to write it
pulsar lily
it's longer than the first way I thought of, but it only uses the second "key point" here
and also that (x^a)^b
and, ab = ba, but that's not about exponents
we're not saying "ok I see a 5 and a 12, maybe I can just multiply them together?"
hopefully that makes sense @twilit tapir
cedar kilnBOT
@twilit tapir Has your question been resolved?
twilit tapir
(1/5)^12 = (5^(-1))^12 = 5^(-12) = (5^12)^(-1) = 1/(5^12)
oh woah
that makes no sense
pulsar lily
take it step by step
twilit tapir
If it was (1/3)^4
would this be 3^4’s reciprocal
pulsar lily
first step: 1/5 is just 5^(-1), so make the substitution to go from (1/5)^12 to (5^(-1))^12
second step: exponent rules say that (5^(-1))^12 is really 5^(-1 * 12) = 5^(-12)
third step: that's fine, but we really want to write -12 as 12 * (-1), so we can write this as (5^12)^(-1)
twilit tapir
Ahhhhh is thee a way you can write it down all the computer writing confuses me
short sand
Hi
pulsar lily
Ahhhhh is thee a way you can write it down all the computer writing confuses me
\begin{align*}
\left(\mathcolor{red}{\frac15}\right)^{12}&=\left(\mathcolor{red}{5^{-1}}\right)^{12}\
\left(5^{\mathcolor{blue}{-1}}\right)^{\mathcolor{blue}{12}}&=5^{\mathcolor{blue}{-1\cdot 12}}\
5^{\mathcolor{blue}{-1\cdot 12}}&=5^{\mathcolor{blue}{12\cdot(-1)}}\
5^{\mathcolor{blue}{12\cdot(-1)}}&=\left(5^{\mathcolor{blue}{12}}\right)^{\mathcolor{blue}{-1}}\
\mathcolor{red}{\left(5^{12}\right)}^{-1}&=\frac1{\mathcolor{red}{5^{12}}}
\end{align*}
wraith daggerBOT
Flip
pulsar lily
better?
pulsar lily
line 3 is using the fact that (-1) * 12 = 12 * (-1)
twilit tapir
Why is there a decimal
pulsar lily
i.e. multiplication is commutative
that's not meant to be a decimal point, that's a centered dot
it's the more common way to write multiplication
twilit tapir
Aokay yes this makes sense
pulsar lily
so it's the product of -1 with 12, equaling the product of 12 with -1
twilit tapir
so I was right about
pulsar lily
yes, but now you can prove it, using only the properties you're given
for each line, make sure you know which property is being invoked on that line
it's important that you're able to follow (replicate and be able to explain to yourself and others) the steps you'd take
twilit tapir
What if the number on the top isn’t 1?
cedar kilnBOT
cedar kilnBOT
✅ Original question: #help-13 message
twilit tapir
yes
pulsar lily
simplify something like $\left(\frac{16}{25}\right)^{\frac12}$
wraith daggerBOT
Flip
twilit tapir
okay so
pulsar lily
good question
where did that 16 run off to?
also wait, where's the original number?
twilit tapir
I don’t know where to place it 😭
pulsar lily
even though I didn't do it on literal paper, I still started with the original expression in my example
I'd like a chain of equalities, like I did
twilit tapir
Okay so I always write the original expression
pulsar lily
yeeeah. it's just easier to follow, crucially so for you if you ever reread your old work
pulsar lily
well you could just copy all the shapes involved in the original expression, and find 16 happily sitting as the numerator of a fraction, wrapped in parentheses
that was probably not what you were asking lol
twilit tapir
it indeed wasn’t 😭
pulsar lily
well, originally we use the fact that 1/x = x^(-1) to proceed, but this time we cannot do that
because it's not a 1 in the numerator, it's a 16
twilit tapir
ooooh wait so it’s something to do with the -1
pulsar lily
is there a way to rewrite the innards of the parentheses in such a way that we at least see an occurrence of a 1/something?
pulsar lily
I want to see the exact rule you're applying and the consequences of invoking it
without the full context, I only see a -1/16, and can confirm that that is indeed a number
pulsar lily
so this is the property you're invoking
how does it apply in this situation? i.e., who is x in our case?
twilit tapir
x is 25
pulsar lily
I was being lazy and colored the things that could be our x
twilit tapir
sorry
pulsar lily
well ok, so what's the consequence? how does the expression change with this substitution?
twilit tapir
It makes the 16 go outside the bracket
pulsar lily
with blahblahblah as the consequence of invoking your property of choice I guess
pulsar lily
it's honestly all the same
however, this equation does not hold
one concern I have is that this looks like $\left(25^{-1}\right)^{\frac12/16}$
wraith daggerBOT
Flip
pulsar lily
what property allowed the 16 in the numerator to traverse through the brackets, all the way to the exponent?
pulsar lily
similar question, what property allowed the 16 in the numerator to traverse into the exponent of the 25?
pulsar lily
since you can't cite a property, that means you aren't allowed to claim these two numbers are equal
that's the rule
I hope that's ok
twilit tapir
oh I’m so bad at maths
I don’t know what any of that meant
CITE A PROPERTY?
WHAT
pulsar lily
you know like citing a source?
twilit tapir
UK education is honestly terrible
twilit tapir
you know like citing a source?
nope
pulsar lily
citing a source is like
pulsar lily
"what property did you invoke?" -> "oh I just used the commutative property of multiplication" or whatever
"oh I used the fact that ab = ba for all real numbers a and b"
twilit tapir
oh I understand
pulsar lily
that's citing a property
pulsar lily
no that's just a toy example
pulsar lily
oh, yeah sure. "by the distributive property, a(b+c) = ab+ac"
this isn't like, sanctioned lingo. I only said "citing a property" because it sounded fitting
twilit tapir
okay okay
pulsar lily
I have no clue
so to rephrase, as a response to this message: because you couldn't point to a property that would have allowed you to claim that the two numbers are indeed equal, you are therefore not allowed to claim that the two numbers are equal
twilit tapir
So what can I do
pulsar lily
you go back and try something else
twilit tapir
wait can we do this once I’ve revised the rest of the content for tomorrows exam
this is extra knowledge for the future
do you mind if you add me
pulsar lily
let me make sure this is all still fair game for an exam setting
twilit tapir
should I show you the questions in the exam
I have the paper
that will be there tomorow
pulsar lily
you can just show a list of properties you're expected to use
twilit tapir
I doubt they’ll go to that far
pulsar lily
do you know how to deal with things like $(2x^2y^{-1})^3$?
wraith daggerBOT
Flip
twilit tapir
no but I have the pages my tutor gave me
I’ll go over them if I struggle I’ll come to you
but I’ll try read over it
Is that fine b
?
If you add me back could we go to dms
when I need to
pulsar lily
sure, that's fine. I'm only asking if you've seen a use of an identity involving $(ab)^n$ on one side
wraith daggerBOT
Flip
pulsar lily
if you genuinely have not, then you're right, it's technically bonus
but if you have, then it could be fair game and worth working on
pulsar lily
ok, so the property in play is that $(ab)^n=a^nb^n$ for all real (nonnegative) $a$ and $b$
wraith daggerBOT
Flip
pulsar lily
and all real n
in fact it holds for all real a,b when n is an integer or rational number whose denominator is not even
but if n is a rational number with even denominator, then it can only be said to work if a and b are both nonnegative; if one is negative, then (ab)^n is not a real number at all, and if both are negative, then (ab)^n is real, while a^n and b^n are not
cedar kilnBOT
@twilit tapir Has your question been resolved?
earnest pecan
Hugh Montgomery (1973)
“The pair correlation of zeros of the zeta function”
Proceedings of Symposia in Pure Mathematics, Vol. 24, AMS.
Anyone know how I can purchase these papers
lean oracle
$\zeta$?!?!?!?!
wraith daggerBOT
Average_X^2
hot cipher
Hugh Montgomery (1973)
“The pair correlation of zeros of the zeta function”
Proc...
This is not a math question tbh, these channels are meant for help on solving problems
lean oracle
WE HAVE Z- oh.
hot cipher
Anyways, look on your dms
lean oracle
me?
earnest pecan
me?
cedar kilnBOT
me?
If you are done with this channel, please mark your problem as solved by typing .close
@earnest pecan Has your question been resolved?
shell echo
any idea on how to find x?
cedar kilnBOT
any idea on how to find x?
sacred anchor
any idea on how to find x?
can you make sure youre not missing any information like last time
shell echo
yeah ive been given the 7cm, 8cm, and 120 degrees
sacred anchor
just checking
shell echo
You can solve this one as is
any pointers on how?
hot cipher
Do you know any trig rules that relate angles and sides?
hot cipher
Well, could you identify which one should be of use?
shell echo
i'm assuming one of these?
hot cipher
Lets go over it
Usually when using these you have to think of "requirements" for you to use them
Meaning, you have to know a set of side lengths and angles to apply the formula such that only one remains unknown
shell echo
for the a^2 one id have to find this right
hot cipher
Yes, there is a problem about that idea
hot cipher
While, yes, you can do that, theres a far easier way
shell echo
what's the easier way?
hot cipher
Look at the right triangle, identify the angles and the length you know
shell echo
by right triangle you mean this one?
hot cipher
yes
hot cipher
You know the three angles and one side
shell echo
can i do
sin(136) / 13 = sin(27,8) / x
hot cipher
Exactly
shell echo
ah okay yeah that makes sense
hot cipher
Why i said this one is easier, apart from the obvious fact that it has less operations involved
The other already required you to find that angle, and then do sine rule to get the length of the left side triangle
To then do cosine
And, as a heads up:
By having 3 specific side lengths that satisfy the triangle inequality you can define one single triangle
having 3 angles, you can define infinite triangles, but they all have the same "shape", so, by knowing a single side, you can pin out all the other sides
And that last one holds too for 2 angles, since you can always find the third.
For some specific situations, 1 could be enough too
shell echo
To then do cosine
yeah true
thanks for all the help and the heads up though, i appreciate it
hot cipher
When you have more sides than angles you should be expecting to use cosine, and viceversa for sine
shell echo
When you have more sides than angles you should be expecting to use cosine, and ...
ah okay true
thank you again
.close
cedar kilnBOT
shell echo
.reopen
cedar kilnBOT
✅ Original question: #help-13 message
shell echo
When you have more sides than angles you should be expecting to use cosine, and ...
i also need help with another problem if you don't mind, this one i kinda have no clue where to start since i missed school when this was taught
Find the areas of the inscribed circle and the circumscribed circle of an equilateral triangle with a side length of 5 cm, and determine the ratio of their areas.
i used chatgpt to translate it so hope it isnt too confusing
shell echo
sure sec
shell echo
should just be this
high bridge
for an equilateral triangle, the centers of both circlesis on the same point, and that is also the center of the triangle
The center of the triangle should be the intersection when you draw heights from all three points of the triangle
high bridge
yeah yeah, something like that
shell echo
alright cool
high bridge
your triangle does not look very equilateral but fine
shell echo
yeah lol sorry
high bridge
the radius of the INSCRIBED circle is the length between the center and the base of a height line
if that makes sense
shell echo
i think i know what you mean
high bridge
ok
and the radius of the CIRCUMSCRIBED circle is the legnth between the center to a point on the triangle
shell echo
ahh okay
high bridge
so um
shell echo
and the radius of the CIRCUMSCRIBED circle is the legnth between the center to a...
so something like this
high bridge
yeah
shell echo
alright
shell echo
the radius of the INSCRIBED circle is the length between the center and the base...
this for inscribed
high bridge
yes something like that
so, a height of an equilateral triangle also bisects the side length and also bisects the angle, you know this right?
shell echo
yes
high bridge
and you also know that all angles in an equilateral triangle is 60 deg
shell echo
yes
high bridge
so, what results from this is:
the radius of Incirc and Circumcirc and 5/2 forms a right triangle with one of the angles being 30 deg
and now you can use trig to solve for the radius of both triangles
shell echo
okay i think i got the general idea of it
bit too late for me though im just gonna head to sleep, i appreciate the help!
.close
cedar kilnBOT
cedar kilnBOT
Available help channel!
primal wigeon
what form is this how do i find its surface
cedar kilnBOT
what form is this how do i find its surface
random swift
cut it up into triangles and rectangles
random swift
maybe like this.
they're both right angle triangles, which are easy to find the area
primal wigeon
i dont have the coordinate M
random swift
nope, you'll need to work that out
white plover
Diff eq question
cedar kilnBOT
Diff eq question
white plover
"after long time has passed" , gives me the idea of steady state solution
which is reliant on the Yp
white plover
yea
white plover
there's two possible solutions
are they C & E?
i was just going to say C and call it a day
random swift
it could be any of them 
blazing dune
I think it means that there is no transient
white plover
isnt that why that part of the graph isnt included
random swift
A if the force is neglidgable compared to the damping factor
B if the force is neglidgable compared to the mass of the system
C if the force is much larger than the damping factor and the mass is relatively small
D if omega is tuned to the resonant frequency of the system
E otherwise
white plover
wats the resonant frequency again
hollow trail
by 'after a long time has passed' i would assume that means that the values of t plotted are long after the force started (i.e. after the transient response has died out)
white plover
by 'after a long time has passed' i would assume that means that the values of t...
yea , so after the oscilitary behavior dies out from the spring
so its just the forced oscilliation
hollow trail
yes
random swift
wats the resonant frequency again
imagine you're on a swing. if you time when you apply a force, you'll increase in height each swing - you've tuned your force to the resonant frequancy of the swing. Do it compleatly out of phase (i.e. apply the force at the bottom of the swing), you'll slow down.
white plover
yea , in that snese
random swift
yeah, C or E in that case
white plover
imagine you're on a swing. if you time when you apply a force, you'll increase i...
ohhhh
hollow trail
resonance means the force is applied at the same frequency as the natural frequency of the system. but that would preclude there being a steady state solution at all
white plover
yeah, C or E in that case
if i had to chose one tho , which is safest
why is this a past test question
hollow trail
itl be like C or E
well how could you get E? there is only one harmonic
white plover
well i dont know what omega value can make it look like that
wait yea
no there is none
random swift
thinking about it a little more, E doesn't seam plausable
hollow trail
E is what you would get if you were to sum two harmonic motions with similar frequencies (beats phenomenon); you might get that with an undamped system if the driving frequency is similar to the natural frequency
but they state here that it's damped
white plover
E is what you would get if you were to sum two harmonic motions with similar fre...
yes this does make sense
thank you both
cedar kilnBOT
@white plover Has your question been resolved?
cedar kilnBOT
light radish
cedar kilnBOT
fossil dawn
cedar kilnBOT
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proper marsh
cedar kilnBOT
livid dirge
have you drawn a diagram to illustrate the problem?
proper marsh
i tried but idk what i was doing
livid dirge
can you show the diagram please?
proper marsh
livid dirge
hint: separate the shadows of the two trees, then join the tip of the shadow of each tree to the top of that same tree.
livid dirge
in your diagram, the shadow of the taller tree covers the shorter tree (and its shadow).
move the taller tree further right, or the shorter tree further left, so that the shadow of the taller tree no longer covers the shorter tree.
(or shorten the shadows of each tree)
proper marsh
livid dirge
... could be further away to avoid confusion, but that works I suppose.
livid dirge
this part. the shadow of the right tree is exactly on the left tree.
(well, its tip is.)
proper marsh
yes
livid dirge
so we can say that the length of the shadow of an object is directly proportional to the height of the object.
look at your figure again. what does this imply about the two triangles?
livid dirge
they are very evidently not equal.
proper marsh
proportional
livid dirge
but you're on the right track. they are equal in some sense.
livid dirge
proportional
you have the spirit of the word, but not exactly correct here.
hint: the word starts with an S.
proper marsh
suplementary
livid dirge
I don't think supplementary triangles exist.
livid dirge
... then why did you say it. 
proper marsh
that means it adds to like 180
proper marsh
... then why did you say it. <:eeveethink:601029006418116608>
divine intelligence
livid dirge
that means it adds to like 180
well, that applies to angles, not triangles.
livid dirge
divine intelligence
then might wanna reconsider which deity you're paying respects to, but anyway...
the word you're looking for is similar. these two triangles are similar because their sides are proportional to each other.
livid dirge
so since you know the triangles are similar, and you know the length of the two shadows, you can easily find the height of the shorter tree via a proportion calculation.
livid dirge
2+x = 5 + 10
there will be some division and multiplication going on.
but there won't be any addition/subtraction.
proper marsh
😢
livid dirge
hint: because the two triangles are similar, the lengths of their sides are related.
consider which side of the shorter tree's triangle is naturally related to the corresponding side of the longer tree's triangle.
proper marsh
2 and 5
proper marsh
idk the equation but i think i found it
because 10/5
which means like every 2 feet tall it is, its 1 foot long shadow
but idk how to solve the x
livid dirge
which means like every 2 feet tall it is, its 1 foot long shadow
you're close. this relationship would help.
do note that your way is not the only way to solve this, though. I was more thinking of comparing the ratios of the pairs of related sides, but yours would work too.
livid dirge
do you want to try completing your method first?
livid dirge
which would be used more in different scenarios
the idea of similar triangles (and thus my way of comparing ratios of related sides) would be more general because sometimes you're given only one pair of related sides and have to find others, but your method is valid in this case too.