what's the question again 😉
#help-10
lone dirge
wary ingot
i want to figure out the chance of getting either combo
lone dirge
A+(B/C/D) OR
B+(C/E/F)
So we should calc the prob of getting none
Either we don't get A and not get B
OR we get A and don't get B or C or D, we know this one already
OR we get B but don't get A or C or E or F
Sounds right?
lone dirge
Fine... Not getting A and B is
wary ingot
that's relatively easy
lone dirge
,w \binom{34}{6}/ \binom{40}{6}
warm shaleBOT
lone dirge
So we are left with the last one
Which we do as we did before...
,w \binom{3}{1} * \binom{40 - 15}{5} / \binom{40}{6} + \binom{3}{2} * \binom{40 - 15}{4} / \binom{40}{6} + \binom{3}{3} * \binom{40 - 15}{3} / \binom{40}{6}
warm shaleBOT
lone dirge
Put everything together and get the compliment
lone dirge
Let's see again...
,w 1 - \binom{34}{6}/ \binom{40}{6} - (\binom{3}{1} * \binom{40 - 15}{5} / \binom{40}{6} + \binom{3}{2} * \binom{40 - 15}{4} / \binom{40}{6} + \binom{3}{3} * \binom{40 - 15}{3} / \binom{40}{6}) - (\binom{3}{1} * \binom{40 - 12}{5} / \binom{40}{6} + \binom{3}{2} * \binom{40 - 12}{4} / \binom{40}{6} + \binom{3}{3} * \binom{40 - 12}{3} / \binom{40}{6})
warm shaleBOT
Requested by Cain#0196
wary ingot
i think we messed up some overlap
lone dirge
,w (\binom{3}{1} * \binom{40 - 15}{5} / \binom{40}{6} + \binom{3}{2} * \binom{40 - 15}{4} / \binom{40}{6} + \binom{3}{3} * \binom{40 - 15}{3} / \binom{40}{6})
warm shaleBOT
lone dirge
That's case 3
Case 1 was:
,w \binom{34}{6}/ \binom{40}{6} - (\binom{3}{1} * \binom{40 - 15}{5} / \binom{40}{6}
warm shaleBOT
lone dirge
and case 2:
,w (\binom{3}{1} * \binom{40 - 12}{5} / \binom{40}{6} + \binom{3}{2} * \binom{40 - 12}{4} / \binom{40}{6} + \binom{3}{3} * \binom{40 - 12}{3} / \binom{40}{6})
warm shaleBOT
lone dirge
,calc 1 - (0.0937 + 0.30886 + 0.05201)
warm shaleBOT
Result:
0.54543
There you go
wary ingot
you kinda lost me
lone dirge
Again
CASE 1: don't get A and not get B
OR CASE 2: we do get A and don't get B or C or D
OR CASE 3: we do get B but don't get A or C or E or F
We want 1 - CASE 1 - CASE 2 - CASE 3
wary ingot
,calc 1 - (0.0937 + 0.30886 + 0.05201)
i mean
we checked each other's answers and they were the same
i'm just suspicious of these new numbers
lone dirge
These aren't new numbers, we had them all before
You added 69.94%
You shouldn't have...
wary ingot
i thought 69.94% was case 2?
lone dirge
So I think that's the answer
Sanity check?
is it less then the sum of the probs?
Yes..
Is it more than each of the probs
yes...
I mean, the sum of the probs of getting the two types of combos
Looks reasonable...
wary ingot
i don't know if i should open another question for that but there's a slightly different case that i don't think i could solve like we did, can we go over that
wary ingot
we were able to solve it because we could isolate A and then isolate B
how would you solve it if you want either
A+B
A+C
B+C
lone dirge
I don't think so...
Because you count A+B+C twice
So we did:
CASE 1: Not A and Not B
Case 2: A and not (B or C)
Case 3: B and not A and not C
wary ingot
so
not A or A+(not B or C)
reversed
+
B+C
does it work if we do that and then subtract A+B+C? 🤔
lone dirge
YES
That's what I wanted to say
Before... The other way of calculating it
Just substract the double counting
Pr(X or Y) = Pr(X) + Pr(Y) - Pr(X and Y)
Generally...
lone dirge
You are welcome 🙂
wary ingot
.close
icy palm
got my two solutions from part a) as -1/2 and 3/2
obtuse pebbleBOT
got my two solutions from part a) as -1/2 and 3/2
icy palm
im not sure how to get all the solutions for b
ik i do cos3x = -1/2 and 2/3
but idk how to get all the possible solution
ik 16.1 is one of them and another is 40
not sure how to get another
ik theres one more
obtuse pebbleBOT
@icy palm Has your question been resolved?
timid silo
Anyone able to help with grade 12 calculus and vectors unit 6 lines and planes
obtuse pebbleBOT
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exotic escarp
obtuse pebbleBOT
exotic escarp
any reason the C got rid of Ln?
polar fossil
<@&268886789983436800>
native inlet
it just looks like log properties were used but I have no clue where the ln went 
polar fossil
no i think they got purged lol
exotic escarp
xD
exotic escarp
it just looks like log properties were used but I have no clue where the ln went...
thats what im wondering
comments are turned off
so Idk what could have happened with natual log
polar fossil
well $\int \frac{1}{x(1-x)}\dd x \neq \ln|\frac{x}{1-x}|$ unless it just happens to
warm shaleBOT
Hayley
polar fossil
sorry what i mean is like
there's the u-sub to worry about
it's not as simple as 1/(...) => ln(...)
exotic escarp
I did partial fractions and u sub for 1/1-x
polar fossil
ah ok that's what I was in the process of doing
native inlet
$\int\frac{1}{x(1-x)}\mathrm{d}x=\ln|x|-\ln|1-x|+C=\ln\left|\frac{x}{1-x}\right|+C$
brisk matrix
probably a typo
polar fossil
yeah that's a typo
exotic escarp
ight well thanks for help
warm shaleBOT
XxMrFancyu2xX
exotic escarp
.close
obtuse pebbleBOT
native inlet
ight well thanks for help
yw! :))
obtuse pebbleBOT
Available help channel!
azure anchor
calculate volume generated by rotation. use any method. i used shell method and am getting a wrong answer but idk where i went wrong.
obtuse pebbleBOT
calculate volume generated by rotation. use any method. i used shell method and ...
polar fossil
shell method would be kinda annoying here
azure anchor
disc instead?
polar fossil
and I'm fairly sure you didn't do it right, the height of the shell would be the right side of the graph minus the left side
yeah do disc/washer
polar fossil
you want to do this in terms of x
polar fossil
(also there's another half of the hyperbola below the axis? but it's symmetric so ig we're ignoring it)
azure anchor
there is, but yeah its symmetric about the x axis so i thought it would be inside the volume anyway
polar fossil
yeah we can ignore it
we would just have to be careful not to double count it
and ignoring it is a great way to make sure of that 
azure anchor
think thats right
just an aside, how could you tell so quickly shell method would be more annoying here than disc?
polar fossil
well first of all i hate shell method in general so maybe i'm biased
but also shell would be something like $\int 2y(2\sqrt{1+y^2}),\dd{y}$
warm shaleBOT
Hayley
polar fossil
so maybe it wouldn't be that bad
yeah that's not bad but still washer just makes more sense to me
obtuse pebbleBOT
low kindle
.reopen
obtuse pebbleBOT
✅
obtuse pebbleBOT
@azure anchor Has your question been resolved?
azure anchor
???
fathom flicker
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gritty goblet
Hey can someone tell me if my linear equation is set up properly for this problem?
obtuse pebbleBOT
Hey can someone tell me if my linear equation is set up properly for this proble...
gritty goblet
My linear equation is 0.34x+0.05(x+74,000)=6892
Cause when I do it it ends up with repeating decimals
Can anyone help me please
.close
merry laurel
A and B are vectors and u is any unit vector
obtuse pebbleBOT
A and B are vectors and u is any unit vector
gilded needle
no, not in general
merry laurel
if A.u = B.u
what i am saying that this is true for every unit vector u
gilded needle
example, suppose A and B are vectors in R^2, and u = (1,0) (the unit vector in the x direction)
oh, for every unit vector u
certainly true in a finite dimensional space
gilded needle
yes definitely true then
just take u = (1,0,0), u = (0,1,0), and u = (0,0,1)
that shows that the x, y, and z components of A and B are equal
therefore A and B are equal
merry laurel
why doesnt this work in the case where u is fixed
gilded needle
if it's just a single unit vector? Then for a counterexample, say we're in R^3 and A = (1,0,0) and B = (1,1,0)
and u = (1,0,0)
then A.u = B.u but A is not the same as B
kind hawk
if you know linear algebra, a unit vector used this way gives a linear map R^3 -> R. that will always have a kernel of size 2
merry laurel
no actually i saw in physics that F.dr = -grad(U).dr for every dr
so it implies F = -grad U
merry laurel
it does not specify every in book
it says lets say we move a distance dr vector
so and so
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
brave oxide
obtuse pebbleBOT
brave oxide
Can someone tell me how the exponent of 6 of the exponential function becomes 3?
zealous dawn
can help me answer this question
vast summit
Can someone tell me how the exponent of 6 of the exponential function becomes 3?
anything square rooted is equalivent to that thing to the power of 1/2 so you have (14)^{1/2} * (e^{6})^{1/2} and when you have powers we multiply so you get e^{3} outside
slate cobalt
2^7
old isle
$(14e^6)^{\frac{1}{2}} = 14^{\frac{1}{2}}(e^6)^{\frac{1}{2}} = e^{6 \frac{1}{2}}*14^{\frac{1}{2}}$
warm shaleBOT
aurelian_s
woven wharf
https://cdn.discordapp.com/attachments/486841085210132490/1119922511119929384/ddda483e-4bb9-4b94-a961-42f72c26f8f1.jpg how doest the marked term came up 0, inside d{.....}
obtuse pebbleBOT
https://cdn.discordapp.com/attachments/486841085210132490/1119922511119929384/dd...
woven wharf
yea but
i didnt get it
how the terms inside d{...}
can u pls solve this on papers
woven wharf
yea i did
polar fossil
you should have +pq at the start and later -pq
woven wharf
ohhh
polar fossil
and the same should be true for all of it
woven wharf
lemme try
i have to expand it again
hold on
i get it
but then why do i have to subtract equation 2 with 1
.close
warped aspen
obtuse pebbleBOT
warped aspen
😭 can someone help me eith 5
desert sinew
For (a), do you remember what is the value of tan 30?
obtuse pebbleBOT
@warped aspen Has your question been resolved?
summer inlet
help how do you solve this
obtuse pebbleBOT
help how do you solve this
summer inlet
both
rich plume
do you know intersecting secants theorem?
summer inlet
nop
obtuse pebbleBOT
@summer inlet Has your question been resolved?
rich plume
This is the intersecting secants theorem
just an application of tangent secant theorem
if you don't want to use intersecting secants theorem you can use tangent secant theorem (That would require a construction tho)
@summer inlet
obtuse pebbleBOT
@summer inlet Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
everyone i've asked says it's a rlly weird question and my working out has gotten me nowhere
obtuse pebbleBOT
everyone i've asked says it's a rlly weird question and my working out has gotte...
pseudo swift
everyone i've asked says it's a rlly weird question and my working out has gotte...
yeah there's a problem with that question
the probabilities don't sum to 1
,w sum for k=0 to infty of (2/3)*(1/3)^(k+1)
warm shaleBOT
obtuse pebbleBOT
@timid silo Has your question been resolved?
pseudo swift
but if we assume the author meant ^(k) instead of ^(k+1) we can work with that
rip @timid silo
obtuse pebbleBOT
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Available help channel!
tender dove
I'm trying to resolve the following limit: $\lim_{x\to+\infty}{2^x\ln x\over x^x}$. I can't really use the order of inifinites because this comes from an exercise that wants me to order infinites. The full exercise asks me to order the following infinites (for $x\to+\infty$): $2^x,x^x,x\ln x,{x^2\over\ln x},2^x\ln x$ and $x^22^x$. I've already gotten that $x\ln x<<{x^2\over\ln x}<<2^x<<x^x$ and $2^x<<2^x\ln x$
obtuse pebbleBOT
I'm trying to resolve the following limit: $\lim_{x\to+\infty}{2^x\ln x\over x^x...
warm shaleBOT
protoleo
tender dove
I get $\lim_{x\to+\infty}{2^x\ln x\over x^x}=\lim_{x\to+\infty}e^{x(\ln2-\ln x)}\ln x$
warm shaleBOT
protoleo
tender dove
I don't really know how to continue
wise briar
i got zero for the limit and wolfram alpha agrees, im not 100% sure if you are allowed but use l'h
dont know why it wouldnt be fine to use l'h though
tender dove
i got zero for the limit and wolfram alpha agrees, im not 100% sure if you are a...
welp... I tried to use l'hopital but the thing becomes huge
wise briar
welp... I tried to use l'hopital but the thing becomes huge
it resolves itself nicely
here let me do that work
tender dove
dont know why it wouldnt be fine to use l'h though
I can use it, I just can't use the ordering of infinites
because I would enter a cyclic definition
wise briar
i get to a point where you have (2/x)^x
and the limit as x approaches infinty of that is 0
which im not sure of the proof of
tender dove
I mean, I do get that from before as well, but wouldn't I get 0 x infinity?
including the other ln x I mean
wise briar
you got this right?
i literally threw the ln(2) out cuz it doesnt matter
wont affect the limit anyways
tender dove
I did get it before
wise briar
then factor out the 2^x and x^x
then if you look at ln(x) + 1/x as x approached infinity you get ln(x) as x approaches infinity and on the bottom its ln(x) +1 as x approaches infinity
tender dove
wait... how does it cancel to 1
wise briar
so you have a +1 difference in the limit so you have (infinity)/(infinity +1) which tends to 1
its just a constant difference
well ig it doesnt "cancel out" but it will tend to 1 as x approaches infinity
tender dove
do I have to cancel out the ln x above and below?
wise briar
no
just think about it
so you basically have an ln(x)/[ln(x) +1]
when you approach infinity
tender dove
I mean, wouldn't I get this? $$1+{1\over x\ln x}\over1+{1\over\ln x}$$
warm shaleBOT
protoleo
wise briar
well yeah
yeah thats the proof ig
both of the fraction terms will tend to zero so youll be left with 1
im kinda just thinking about the problem im not really trying to prove it
tender dove
Ok, for the other factor I basically have already demonstrated it while doing the the other confrontations, so I guess we are done
wise briar
you got the (2/x)^x ?
tender dove
Thank you very much!
wise briar
yep
tender dove
you got the (2/x)^x ?
yep
wise briar
yw
tender dove
you got the (2/x)^x ?
I get 0^+infinity, but anything between -1 and 1 elevated to a big number goes to 0
so that's not really an indefinite form
tender dove
Once again, thank you! ^^
obtuse pebbleBOT
obtuse pebbleBOT
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daring fiber
Hey
daring fiber
find surface boundried by y=x,y=2x and xy=1
This graph
So this question is from calc 1
A friend of mine has it
But i am finishing calc 3 and basically don’t remember good how we used to do these
But do we seperate this in 2 different integrals
brave bramble
Ye
daring fiber
So give me a sec and if you can check if these are right
Is this right?
Or am i messing it up
Yh i have completely forgotten them after the double integrals and volumes
Wait this
Looks better
Anybody can confirm if this is correct
obtuse pebbleBOT
@daring fiber Has your question been resolved?
obtuse pebbleBOT
@daring fiber Has your question been resolved?
left jetty
How would i find the maximum height of the arch in 43 c) ? I dont know where to go from here
obtuse pebbleBOT
How would i find the maximum height of the arch in 43 c) ? I dont know where to ...
left jetty
<@&286206848099549185>
kindred plinth
have you found the equation of the slope?
obtuse pebbleBOT
@left jetty Has your question been resolved?
left jetty
have you found the equation of the slope?
yeah y prime in my work
left jetty
do you know what is implied when someone says the slope of a curve is 0?
its a local max or min
i set the slope to 0 above but idk how to simplify the equation to get the x values
kindred plinth
you can start by getting rid of the -0.345 since the left hand side is 0
then it will be little easier to calculate
left jetty
then it will be little easier to calculate
ok i got rid of the -0.345 im still a bit confused about what to do next
kindred plinth
can you show me your work
left jetty
can you show me your work
kindred plinth
okay on the right track
what do you think you should do next? make any guess
keep in mind you have a -ve sign and 0 on the other side
left jetty
what do you think you should do next? make any guess
i think maybe factor out the e^0.0329x?
kindred plinth
that would be right but that would little complicate the solution process
there is another easier way think if you can
left jetty
there is another easier way think if you can
im not sure what other way i could do
kindred plinth
its okay
you can bring the -e^ thing to your left hand side, by adding e^0.0329x both side
left jetty
you can bring the -e^ thing to your left hand side, by adding e^0.0329x both sid...
wouldnt it be e^-0.0329x?
kindred plinth
yes
then you have +ve e on both side, then you can compare the exponents if you know what i mean
left jetty
then you have +ve e on both side, then you can compare the exponents if you know...
so then it would be -0.0329x = 0.0329x?
kindred plinth
so what do you think x is from here?
left jetty
so what do you think x is from here?
x=0?
kindred plinth
yes you are absolutely right
so now you have found the value of x, from here you can find y
mossy gyro
obtuse pebbleBOT
obtuse pebbleBOT
@mossy gyro Has your question been resolved?
full cloud
$dl = rd\theta = 2a\sin\theta d\theta$
warm shaleBOT
lmaowhat
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
hi i need help with this equation
obtuse pebbleBOT
hi i need help with this equation
keen cedar
Are you asking about number 8?
timid silo
yes
keen cedar
Okay, so you want to find a common denominator for the two fractions. Which means you want to have the same numbers on the bottom for both fractions. One hint is that (2m+3)/(2m+3)=1 so you can multiply a fraction by (2m+3)/(2m+3) and it is still the same fraction, because you are just multiplying by one. Give that a try on the left fraction.
timid silo
i dont understand
keen cedar
Okay, I'll send a pic to help. one sec.
timid silo
ok
keen cedar
timid silo
oh yes i understand
keen cedar
We know (2m+3)/(2m+3) = 1 since any number divided by itself is 1. Like 5/5 = 1 , 100/100 = 1, 2x/2x = 1 and so on
timid silo
we an fraction the top and bottom are the same value it is equal to 1
keen cedar
Yes, exactly
timid silo
ok i understand ty
keen cedar
No problem, good luck!
turbid maple
would the width be 3x+7?
and the length 5x?
obtuse pebbleBOT
would the width be 3x+7?
and the length 5x?
high lily
assuming this is supposed to be a rectangle, yes
earnest phoenix
fixed
correct if you're going to use it for calculation, but don't confuse the concepts of width and length
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
clear sonnet
obtuse pebbleBOT
clear sonnet
How would you integrate this?
I tried doing polynomial long division
but that didn't work out
is this just partial fraction decomposition?
gilded needle
what if you write $x^3 + 4 = x(x^2 + 2) - 2x + 4$
warm shaleBOT
Bungo
gilded needle
or equivalently, carry out the division
gilded needle
yea you can do it that way
clear sonnet
I tried doing that but it didn't work out
gilded needle
i just mentally said, i have an x^3 but it would be nice if i instead had x^3 + 2x, because that has x^2 + 2 as a factor.. so I just add and subtract 2x to get what i want
gilded needle
that last line should be -2x + 2 (you forgot to bring down the +2)
also, it should have been x^3 + 4 not x^3 + 2 to start with
clear sonnet
right
gilded needle
so after making those corrections, you get -2x + 4 on the last line
that's your remainder
clear sonnet
gilded needle
so the result is x + (-2x+4)/(x^3 + 4)
gilded needle
should be +4 on the last line not -4
clear sonnet
yeah
gilded needle
so yea that's equivalent to what i did
gilded needle
sure
obtuse pebbleBOT
Available help channel!
static mango
Hello, I have this formula for finding the temperature of a system given certain parameters. However, I'm not sure how to solve this when you have two T values and one of them is to the power of Euler's constant. It is a mess that I have been unable to detangle and would appreciate advice on.
obtuse pebbleBOT
Hello, I have this formula for finding the temperature of a system given certain...
obtuse pebbleBOT
@static mango Has your question been resolved?
polar fossil
i would be rather surprised if this has a nice solution
tardy epoch
Hello, I have this formula for finding the temperature of a system given certain...
You're probably supposed to Taylor expand the exponential function
static mango
How would you Taylor expand this?
tardy epoch
,tex .maclaurin
warm shaleBOT
rie.mann
tardy epoch
First row
static mango
alright
static mango
.close
wide tendon
obtuse pebbleBOT
brisk matrix
!status
obtuse pebbleBOT
What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above
@wide tendon Has your question been resolved?
wide tendon
.close
vestal marten
How many rectangles can be made in a n × m grid of points?
Example: 2 × 3 grid.
1 2 3
1 ● ● ●
2 ● ● ●
There are 3 rectangles:
{ (1,1), (1,2), (2,1), (2,2) }
{ (1,1), (1,3), (2,1), (2,3) },
{ (1,2), (1,3), (2,2), (2,3) }
Each rectangle is uniquely determined by the set of its 4 points (which i represent as (line, column) pairs).
obtuse pebbleBOT
How many rectangles can be made in a n × m grid of points?
Example: 2 × 3 grid....
thin pawn
BlackPenRedPen made a video on that iirc. Maybe that could help you out
polar fossil
figuring it out is a fun exercise though!
obtuse pebbleBOT
@vestal marten Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
frozen kindle
part b and part c
obtuse pebbleBOT
part b and part c
unkempt bramble
!show
obtuse pebbleBOT
Show your work, and if possible, explain where you are stuck.
forest sinew
wait but theres 8 people
what do they mean someone has to use a forward facing seat
theres 8 people
unkempt bramble
Some PPL stand for exercise maybe
obtuse pebbleBOT
@frozen kindle Has your question been resolved?
polar fossil
what do they mean someone has to use a forward facing seat
I've seen this problem before. I came to the conclusion that they must mean that one specific person has to sit in a forward facing seat
forest sinew
I've seen this problem before. I came to the conclusion that they must mean that...
OH
polar fossil
I didn't consider the possibility of aisle surfing to be fair
shell ether
I hope I'm in the right place.
Hello, I'm PolyBytes and I'm new 
.
To make a long story short, I am a video game programmer and I love seeing what sorts of amazing things can be implemented in video games using mathematics. I am currently hyperfixating on the topic of Catenarys. Catenarys describe the way a rope, chain, cable, etc. that is suspended from both ends will droop under the influence of gravity. I am trying to understand how the math behind the equations for catenarys works so that I can then translate that into an algorithm to create a suspended cable in a video game I am working on.
My issue is that I get confused easily when it comes to math that goes much beyond trigonometry and algebra. I don't have very much experience with subjects such as calculus for example. I may be lacking the necessary knowledge to tackle this sort of problem and if so I would like some direction as to how I can expand my knowledge of mathematics to the point where this is no longer unapproachable for me.
Ultimately I want to understand the equations for the catenary and as of now I am unsure what my roadblock is. It may be that I just don't understand certain mathematical terminology or perhaps I'm lacking knowledge in a specific area of mathematics that is preventing me from wrapping my head around this.
Thank you for reading.
Catenary wikipedia article: https://en.wikipedia.org/wiki/Catenary
obtuse pebbleBOT
I hope I'm in the right place.
Hello, I'm PolyBytes and I'm new <a:s_wave:81665...
polar fossil
you can approximate a catenary with a bunch of small pieces that all have weight and pull on each other (kind of like how a chain itself works)
that plays well with physics engines if you want to make it swing or manipulate it and have it fall back down to rest
royal basin
into an algorithm to create a suspended cable in a video game I am working on.
do you want it to be static or to like, follow physics
obtuse pebbleBOT
@shell ether Has your question been resolved?
shell ether
> into an algorithm to create a suspended cable in a video game I am working on....
static, calculated once and then not again
royal basin
right...
i mean ok like, properly deriving the equation of the catenary takes quite a lot of work with like. calculus of variations (which is a pretty beefy subject with at least diffeqs as prerequisite) and shit
shell ether
you can approximate a catenary with a bunch of small pieces that all have weight...
that would work for just a few catenarys but the scale of this is much larger unfortunately and all I need are static catenarys that are calculated once and then not touched again
polar fossil
yeah understanding those equations is hard, you're probably best off just implementing the curve as a function
shell ether
i mean ok like, properly _deriving_ the equation of the catenary takes quite a l...
what would you recommend that I do to be able to wrap my head around this?
royal basin
learn calculus maybe
but as far as implementation goes you will need to somehow use the 'general' equation of a catenary, which is y = a cosh(x/a)
where cosh(t) = (e^t + e^(-t))/2
shell ether
I actually got as far as y = a cosh(x/a) but the math breaks down when the two end points are of different y values or different distances from zero on the x axis
the part I don't understand is cosh(t) = (e^t + e^(-t)) / 2
royal basin
that's the definition of the cosh function
anyway like
further details would be implementation dependent
so i dont really want to say anything lest it end up not fitting
obtuse pebbleBOT
@shell ether Has your question been resolved?
azure anchor
what engine are you using
most capable engines have physics capabilities built in that should do the heavy lifting for you
if you must understand the inner workings, i would suggest looking into the definitions of the functions pertinent to whatever it is you're trying to design/implement.
obtuse pebbleBOT
@shell ether Has your question been resolved?
azure anchor
you haven't even provided if you're working in 2 or 3 dimensions.
obtuse pebbleBOT
@shell ether Has your question been resolved?
worthy cargo
problem: Prove that if a is congruent to b (mod x) and y | x, then a is congruent to b (mod y)
obtuse pebbleBOT
problem: Prove that if a is congruent to b (mod x) and y | x, then a is congruen...
worthy cargo
so for my approach I let
a - b = xk for some integer k
then I divided both sides by y to get
(a-b)/y = x/y * k and since y | x, x/y is an integer and so is k
so since (a-b)/y is an integer, a is congruent to b mod y
is that correct
obtuse pebbleBOT
@worthy cargo Has your question been resolved?
obtuse pebbleBOT
@worthy cargo Has your question been resolved?
worthy cargo
<@&286206848099549185> let me know if there is any flaw in my reasoning
sage geode
Looks good
frigid pulsar
“Solve”
candid yarrow
Can someone tell me if this is a combination
what is the context?
frigid pulsar
what is the context?
I’m not even sure myself, it just says to solve
I think it’s a combination but I’m not quite sure
candid yarrow
are you learning combinations?
candid yarrow
Is this exam on combinations?
candid yarrow
what other possible meanings are there for this have you been taught?
frigid pulsar
I forgot but I think it had something to do with a binomial formula
I’ll just check the book afterwards
Thanks though
obtuse pebbleBOT
@frigid pulsar Has your question been resolved?
candid yarrow
I forgot but I think it had something to do with a binomial formula
that should also be a combination
frozen kindle
idont know how to approach part d
obtuse pebbleBOT
idont know how to approach part d
candid yarrow
i tried doing 5! x 4!
why did you do that?
frozen kindle
y can be arranged in 5 positions of the word
and then the remaining 4 letters are factorial?
i honestly hv no idea
candid yarrow
You could add the ones that begin with Y to the ones that have Y as their second letter, then to those that have Y as their third letter and so on
Can you justify for yourself that you haven’t double-counted and that the number of elements in each of these sets is the same?
candid yarrow
idont know how to approach part d
Does this help?
frozen kindle
one sec
so if it were in the first position
then it would be 7P4
equal to 840
and then 5 more times
which is equal to 4200??
candid yarrow
this is right
frozen kindle
ohhhhh
candid yarrow
Can you justify for yourself that you haven’t double-counted and that the number...
Can you see the answer to this question?
candid yarrow
If you’re interested, you can prove that there isn’t any overlap between the words that have Y in the first place and the words that have it in the second place, and that there are the same number of each
answer:
||because there’s only 1 Y, so it can’t be in the second and first place at the same time||
||because there are the same number of five letter words that have Y in the nth place as there are four letter words that don’t have Y, so there are the same number of five letter words with Y in the nth place as there are five letter words with Y in the mth place, where n and m are both less than or equal to 5||
This is just to ensure that the calculation didn’t overcount or undercount
Since the answer key says 4200, you’re right anyway so you don’t need any of this
elfin vapor
obtuse pebbleBOT
elfin vapor
just a quick one, is this a correct force diagram. as it means horizontal component is the same as the parallel component (both sin(theta) , is that right? and then vertical and perpendicular are the same?
obtuse pebbleBOT
@elfin vapor Has your question been resolved?
woven mantle
obtuse pebbleBOT
woven mantle
I’m not sure if my answer is correct or how to get the right one
nocturne minnow
You never answered what I asked before
nocturne minnow
I asked you how did you get $y = \frac{-3}{-2}x - 7$
warm shaleBOT
dldh06
obtuse pebbleBOT
@woven mantle Has your question been resolved?
woven mantle
I asked you how did you get $y = \frac{-3}{-2}x - 7$
I graphed it with the slope from the 4,-2 point and found the y intercept to be 7
nocturne minnow
I graphed it with the slope from the 4,-2 point and found the y intercept to be ...
Can you mathematically find the equation of the line given that info?
@woven mantle It is not helping if you take ages to reply back. Here is a video that you can use, the first half is parallel equations, the second half is perpendicular.
https://www.youtube.com/watch?v=LTb2-LE7StE&ab_channel=TheOrganicChemistryTutor
Here is one that is similar to your problem
https://www.youtube.com/watch?v=u8YBdN23rK4&ab_channel=Mario'sMathTutoring
mossy gyro
is (x-a)² = (a-x)²
obtuse pebbleBOT
is (x-a)² = (a-x)²
spring trail
!help
obtuse pebbleBOT
Please read #❓how-to-get-help
spring trail
is (x-a)² = (a-x)²
why dont u expand and see?
(3-2)squared and (2-3)squared
or compare this
spring trail
hmmmm
mossy gyro
is (x-a)² = (a-x)²
But this is false know?
spring trail
nope
ruby path
no it's true
mossy gyro
Oh thanks
mossy gyro
Due to double and triple integral I'm forgetting simple algebra
obtuse pebbleBOT
@mossy gyro Has your question been resolved?
timid silo
If I were to sleep one hour less, how much % more do I have of the year?
grizzled shore
What have you tried
timid silo
Like if I sleep one hour less for one day, how much time (in percent) would I save of the year?
1/24 is 4.6%
It's 4.6% percent of the day
grizzled shore
Let’s consider that sleeping 1 hour less is relative
If you normally sleep 24 hours a day, sleeping 1 hour less is an “infinite” gain in time
timid silo
Oh
grizzled shore
Whereas if you normally sleep 1 hour a day, sleeping 1 hour less is a relatively small gain
grizzled shore
Yeah you want to know what’s the base amount of time you have
Then calculate what % increase is having 1 more hour?
Then since every day in a year has 24 hours, finding the % for a day is the same as finding the % for a year
timid silo
29.2% of my day consists of sleep, considering I sleep 7 hours on average
grizzled shore
So normally you have 17 hours in a day
timid silo
If I were to reduce 1 hour which is 4.6% I would get to 24.6% total
grizzled shore
If you sleep 1 less hour you now have 18 hours in a day
timid silo
If you sleep 1 less hour you now have 18 hours in a day
Which should be 75.4%
grizzled shore
So what % difference is 17 vs 18 hours in a day
grizzled shore
No that’s not how it works
If I normally have 1 hour in a day
And I suddenly now have 2 hours in a day
My time has increased by 100%
ie I’ve doubled the amount of time I have
grizzled shore
Yes
timid silo
So in a year, if I were to sleep everyday exactly 7 hours a day compared to 8 hours a day I would have a total of 5.9% more time compared to before
timid silo
Oh yes
grizzled shore
You’ll have 5.9% more time
grizzled shore
The same
timid silo
I would have to do the same thing but now with 365
grizzled shore
Then since every day in a year has 24 hours, finding the % for a day is the same...
.
timid silo
I mean if I sleep 7 hours everyday for 365 days, but on one day I would sleep only 6 hours, how much extra time would I have?
grizzled shore
If I sleep 8 hours a day (a third of the day) then I sleep a third of the year
grizzled shore
I mean if I sleep 7 hours everyday for 365 days, but on one day I would sleep on...
That’s just 1 hour a day * 365 days
= 365 hours
timid silo
Oh right
Thank U very much
I appreciate your help
And Ur time
Now I have an excuse in order not to complain about not getting enough sleep
I save more time
obtuse pebbleBOT
Available help channel!
golden whale
Hey can someone help with this
obtuse pebbleBOT
Hey can someone help with this
golden whale
obtuse pebbleBOT
@golden whale Has your question been resolved?
obtuse pebbleBOT
@golden whale Has your question been resolved?
polar fossil
looks like you have the right answer
wolfram just did some simplification and factoring
fading wadi
I have this equation and I need to convert it to an equivalet equation with just 0 on the right side snd find the x intercepts. I’m not sure how to deal with it since there are 2 absolute value sets. |2x-5|=4-|x-3|
obtuse pebbleBOT
I have this equation and I need to convert it to an equivalet equation with just...
frosty river
You have to define branches for the two absolute values simultaneously
2x-5=0 → x=5/2
x-3=0 → x=3
So you have to study the sign of 2x-5 and x-3 in these intervals (-∞,5/2), (5/2 3), (3,∞)
fading wadi
how about the other side
frosty river
You get three different equations
fading wadi
so i just plug in numbers into the equation
from those intervals
and check wether they’re positive or negative?
frosty river
Yes. Then making the expressions positive with the proper negative signs you get three different first order equations which you can solve
fading wadi
wait what
im so confused
what do you even mean positive or negative and then how do I make 3 diff equations
frosty river
Have you ever writen an absolute value with two branches?
frosty river
For |x-3| you have that
|x-3| = -(x-3) when x is in (-∞, 3)
|x-3| = x-3 when x is in (3,∞)
fading wadi
Ok
so what do you do with thst
that
can you solve this and send me a pic or smth
<@&286206848099549185>
obtuse pebbleBOT
@fading wadi Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
obtuse pebbleBOT
uneven palm
do sin(240) / cos(240)
high lily
your reference angle is wrong
timid silo
270 - 240 = 30?
high lily
why are you subtracting from 270
timid silo
thats how i learned to do it and it worked for me until now
high lily
the reference/ related acute angle is the acute angle between the terminal side and the x-axis (not y-axis)
polar prism
obtuse pebbleBOT
polar prism
polar prism
btw A=0, B=1 and C=2
obtuse pebbleBOT
@polar prism Has your question been resolved?
obtuse pebbleBOT
@polar prism Has your question been resolved?
obtuse pebbleBOT
violet haven
I need help\
obtuse pebbleBOT
@iron perch Has your question been resolved?
obtuse pebbleBOT
@iron perch Has your question been resolved?
candid yarrow
I need help\
please read #❓how-to-get-help
obtuse pebbleBOT
@iron perch Has your question been resolved?
obtuse pebbleBOT
@iron perch Has your question been resolved?
opaque galleon
For absolute convergence test, do I have to check that $\sum_{n=1}^{\infty} a_n$ is convergent or can i go straight to $\sum_{n=1}^{\infty} |a_n|$
obtuse pebbleBOT
For absolute convergence test, do I have to check that $\sum_{n=1}^{\infty} a_n$...
warm shaleBOT
casiofx991exz
opaque galleon
I dont have to right
polar fossil
isn't that the point of the absolute convergence test? to tell you whether the original series converges?
opaque galleon
isn't that the point of the absolute convergence test? to tell you whether the o...
yeah if absolutely convergent then convergent
but if it's divergent
I guess I have to check a_n
polar fossil
yeah if it's divergent it doesn't tell you anything useful
but if a_n diverges then so does |a_n|
for what that's worth
opaque galleon
yeah but if a_n is convergent and |a_n| is divergent then it's conditionally convergent
polar fossil
yerp
obtuse pebbleBOT
@opaque galleon Has your question been resolved?
obtuse pebbleBOT
brave bramble
Simplify the nested fraction
One easy way here is to multiply top and bottom by 2+h
polar fossil
yes
warm frigate
obtuse pebbleBOT
warm frigate
how do i differentiate this please
i know the 5 will dissapear
and x will become 1
but the fraction what do i do?
polar fossil
rewrite it as $(x-2)^\inv$
warm shaleBOT
Hayley
polar fossil
now can you differentiate it?
graceful kestrel
polar fossil
yeah that works too
warm frigate
**Hayley**
Wait so if i wrote it like this do i just subtract the power
which will be (x-2)^-2
or am i using chain rule?
polar fossil
yep (make sure to multiply by the power first)
you can use chain rule if you want but $\dv{x}(x-2) = 1$ so
warm shaleBOT
Hayley
polar fossil
it won't change much
warm frigate
ohh alright
so the final answer will be 1 - -1(x-2)^-2
wrote it here in fraction form
obtuse pebbleBOT
warm frigate
have a good day
obtuse pebbleBOT
Available help channel!
knotty pendant
for t formula can u let t be anything like tanx or tan2x or 2tanx
obtuse pebbleBOT
for t formula can u let t be anything like tanx or tan2x or 2tanx
@knotty pendant Has your question been resolved?
obtuse pebbleBOT
Available help channel!
vernal kayak
I don’t understand this (I’m supposed to learn this by myself). I’m gonna separate a paragraph for each question I have I guess.
I understand how if you have to add it, subtract it or multiply it, you just put it in the function and do it. But for division, I don’t understand the domain restriction.
I also don’t understand what g(x) or h(x) even is Also, whenever they say something like g+h, you have to plug in g(x)’s function for g and h(x)’s function for h? Why do they add the “(x)” in the equation but not in the operation?
I know that f(g(x)) = h(x). On this page, anytime there is just a letter (like f), it means it is substituting the equation (f would be substituting the equation of f(x) which is -4x+3). Because f(g(x)) = h(x), if you plug the equations provided in, they don’t equal. I understand f is outer function and g is inner function but I’m confused as to how they work or what they are even supposed to do.
I also awnsered everything exept for the last part where is said identify the domains of each function, I don’t understand what to awnser there.
obtuse pebbleBOT
I don’t understand this (I’m supposed to learn this by myself). I’m gonna separa...
vernal kayak
willow ravine
What do you mean you don’t know what g(x) or h(x) “even is”?
vernal kayak
Other than g is inner function and h(x) = f(g(x)), I don’t know anything about them. I don’t know how they are used in this context.
Like what is h(x) what does it do.
willow ravine
It’s just a name for a distinct function.
vernal kayak
What is a distinct function
willow ravine
It’s like if someone’s pay is calculated by PAY = 15 + x (where x is number of years at the company) and taxes is calculated by TAX= PAY * x (where here x is the % based on their yearly income)
PAY and TAX is two different distinct functions, but an entirely different function h(x) = PAY - (PAY * TAX)
All of those functions are “distinct” because they have specific answers for values of x and they don’t equal each other for all x.
Where do you see h(x) = f(g(x))?
vernal kayak
Hold on I was doing something else and I’m gonna process what you just said.
Your explanation makes so much sence. So PAY would be like g(x) and TAX would be like f(x) and h(x) uses them together. So if they are all distinct functions (so they have different x), how can you add them together. Adding them doesn’t make sence.
vernal kayak
Where do you see h(x) = f(g(x))?
In my notes, it was from composition of functions.
willow ravine
Yeah, but in your problem you’re not using composition. In your picture.
willow ravine
Your explanation makes so much sence. So PAY would be like g(x) and TAX would be...
Let’s say you define a function s(x) ( _(x) is just a way to name a function that changes value based on x).
Let s(x) be the total sun the company pays out per hour. So you would add all the PAY functions of each employee, let’s say ALICE (a), BILL (b), CHARLES (c).
They all have distinct PAY functions, where x is their number of years at the company.
But, s(x) = a(x) + b(x) + c(x)
Where, a(x), b(x), c(x) is their pay.
obtuse pebbleBOT
@vernal kayak Has your question been resolved?
vernal kayak
Thank you.
You cleared it up a lot
However, I still don’t know how to do the last part. Where is asks identify the domains of each function (and everything below)
obtuse pebbleBOT
@vernal kayak Has your question been resolved?
hexed ravine
obtuse pebbleBOT
hexed ravine
hexed ravine
this is my answer^
hexed ravine
This is the question
this is the correct answer^
I'm assuming I made my diagram wrong and my angle is incorrect
but I can't understand how to fix it if that's the case
apologies for my messy handwriting
any help is appreciated!
obtuse pebbleBOT
@hexed ravine Has your question been resolved?
glossy basalt
hi, i think you made a mistake on N25°E
N25°E means:
Pointing North, but tilt 25° due East.
glossy basalt
so would that outer angle be 25° and the inner angle be 65°?
yes
hexed ravine
let me try it one second
I didn’t change the diagram because it would take too long, but the inner angle of the triangle on the right comes out to 155
which is too large considering the winds heading in 41°
@glossy basalt
hexed ravine
Ahhhh okay, I think I over complicated it
glossy basalt
note that
v is what we need to find
and
v+w = (500km/h, N25°E)
hexed ravine
aren't we supposed to find the resultant?
glossy basalt
the destination of the plane is N25°E, we need to find a velocity of the plane that even with the strong wind, it can reach the destination
hexed ravine
That’s the ground speed?
glossy basalt
since the airspeed (which is the final speed) is 500km/h, and the wind is somewhat pushing it behind, we'll have to get a bit more than 500km/h
argh
sorry
I can't explain well
hexed ravine
Nono I understand
I understand the concept and what we need to get to, I just don’t know how to manipulate the angles and the vectors to get an angle that works
I know how to get the ground velocity using cos law, and how to get the angle the plane is moving using sin law
But I just don’t know how to apply them
hexed ravine
Alrighty thank you
glossy basalt
i think i messed up the airspeed and groundspeed
(500km/h, N36°E) + (100km/h, S41°E) = (532.3km/h, N25°E)
glossy basalt
(500km/h, N36°E) + (100km/h, S41°E) = (532.3km/h, N25°E)
would this make sense to you?
@hexed ravine
hexed ravine
Yea
except the wind is blowing from S41°E not towards it I think
so it would be N41°W right?
glossy basalt
yea
hexed ravine
I think I just need to get the angle between w and v
and use that to get the ground velocity and the angle the plane needs to travel
but I'm having trouble getting it
glossy basalt
sorry i was on my way back home just now. I'm home now, gotta take a quick shower and continue with the question
obtuse pebbleBOT
@hexed ravine Has your question been resolved?
glossy basalt
hexed ravine
I was trying to do it similar to this example
but that's probably not how you get it lol
glossy basalt
glossy basalt
oh i see
glossy basalt
this Q5 is easier
glossy basalt
this Q9 is far more tricky
the question is asking 2 different components of different vectors
we'll have to first identify which one is which and the use tip-to-tail method to find the triangle
do tell if you have any question up till here
glossy basalt
A pilot wishes to fly......N25°E of his present position.
(this is about the ground velocity)
...an airspeed of 500km/h...
(this is about the air velocity)
...determine the heading the pilot should steer...
(this is about the air velocity)
... the ground speed ....
(this is about the ground velocity)
so, we'll have to find
- the ground speed
and - the air steer
hexed ravine
OHHHHH so the ground speed has the heading of N25°E?
glossy basalt
yea
hexed ravine
I thought the airspeed was the one with the direction
glossy basalt
nah
hexed ravine
k sorry that makes sense
hexed ravine
so we would have to find the air steer first then?
and then use that to find the ground velocity right?
glossy basalt
that'll be the easier way, so ya
glossy basalt
so the angle is 180-(66+90) right?
where did the 90° come from?
hexed ravine
so can't we use sin law to find it then?
glossy basalt
we can
glossy basalt
so the angle is 180-(66+90) right?
just a little bit more complex than this
glossy basalt
you can do that too
hexed ravine
but that gives me an angle of 11°
hexed ravine
and the correct one is 36
glossy basalt
which is also correct
hexed ravine
is it 25+11?
like the 25° from the ground velocity + the 11° from the angle between the ground velocity and the air speed?
glossy basalt
correct