help-26

[toxic aspen]

$P(9,5) = \frac{9!}{(9-5)!} = \frac{9!}{4!} = 98765 = 15120$ ?

[thorny flame]

Merineth

[toxic aspen]

Am i thinking correct?

[coarse tusk]

Why are you doing P(9, 5)

[toxic aspen]

Well the total number of letters are 9 in MATEMATIK

r = 5 i got since we have letters that are the same

MMATEATIK and MMATEATIK are the same

[coarse tusk]

Why 5 though

There are two M’s and two A’s and two T’s

[coarse tusk]

Yes precisely

Okay let me ask you this

[toxic aspen]

M, A, T, E, I ,K

woops

[coarse tusk]

How many ways of arranging the letters would there be if all the letters were distinct

[toxic aspen]

should it be 6?

[coarse tusk]

There are 6 unique letters yes

P(9, 6) is not the right answer though

[toxic aspen]

Hmm

I thought r was the amount of letters i'm arranging?

and n is the total?

[toxic aspen]

9! ?

[coarse tusk]

Yes

Now just divide by the number of duplicates you can get

[coarse tusk]

Use this logic

[toxic aspen]

If every letter is distinct then r is 9

I thnk?

Seems odd since it'll divide by 0

I don't know \o/

[ruby tree]

0! =/= 0

[toxic aspen]

OH RIGHT

0! = 1?

[ruby tree]

Yes but that's besides the point

Simpler problem: the letters are MMAT

[toxic aspen]

Like if someone asked me to arrange the word word then i'd do $P(4,4) = \frac{4!}{(4-4)!} = 432'1 = 24$

[thorny flame]

Merineth

[coarse tusk]

Even if you have duplicate letters, you’re still arranging 9 letters

It’s just that those arrangements will have duplicate values

[ruby tree]

I mean sure but you don't need permutations or combinations if all the letters are distinct

[ruby tree]

4! ways if the two Ms are distinct, right?

[toxic aspen]

Yes

[ruby tree]

But the two Ms aren't distinct, so

If you think of any possible ordering, you would find that there are two ways among the 4! that give that same ordering

Simply because you can swap the Ms

So instead of 4!, what would be the total without duplicates?

[toxic aspen]

I'm not sure but gpt says that i should divide with the total amount of duplicates? In this case 2!? Since we have 2 M's

[ruby tree]

List them and count them

[cedar wagon]

Gpt will replace all of us

[toxic aspen]

That's the all distinct arrangements for the word MMAT

[ruby tree]

Did you really ask GPT to do that for you?

[toxic aspen]

Yea, was faster

[ruby tree]

[toxic aspen]

MMAT
MMTA
MAMT
MATM
MTAM
MTMA
AMMT
AMTM
ATMM
TAMM
TMAM
TMMA
MMAT (repeated)
MMTA (repeated)
MAMT (repeated)
MATM (repeated)
MTAM (repeated)
MTMA (repeated)
AMMT (repeated)
AMTM (repeated)
ATMM (repeated)
TAMM (repeated)
TMAM (repeated)
TMMA (repeated)

Is there a reason why wer can't use the formula for Permutations?

Yeah we have covered that

$\frac{9!}{2!2!2!}$

[thorny flame]

Merineth

[toxic aspen]

This is the solution given for the word MATEMATIKbut the formula for permutations P(n,r) = n!/(n-r)! only uses one divisor in the denominator, why?

Yes i see that. But why?

So i'm just meant to memorize the formula without understanding why?

Why do we divide by 2x2x2?

[regal harbor]

u could watch khan academy

https://www.youtube.com/watch?v=DROZVHObeko&pp=ygUZa2hhbiBhY2FkZW15IHBlcm11dGF0aW9ucw%3D%3D

[toxic aspen]

I've already watched it

[keen linden]

idk i can't explain

[toxic aspen]

https://youtu.be/XJnIdRXUi7A?t=288 this guys explains it as n is the total amount of letters and r is how many i'm choosing from

Why in my case isn't MATEMATIK n = 9 since we have 9 letters and 6 since we are choosing from 6 distinct letters being M, A, T, E, I, K ?

[regal harbor]

have you watched the missisipi example

[regal harbor]

you'll get this

[toxic aspen]

Yeah i have but i still dont get it

He isn't using the formula for permutations

[regal harbor]

sometimes you have to take it as it is

[toxic aspen]

Okay so the formula for permutations isn't definitive? I just have to memorize it instead?

[regal harbor]

the khan academy explanation was sufficient for me

[toxic aspen]

Wasn't sufficient for me since i'm clearly tarded

[regal harbor]

it's ok

we're all stupid when it comes to math

[toxic aspen]

All the examples are dogshit because they always vary

there isn't like a clear "This is how it's done and why"

Every example gives something new to it

[regal harbor]

just do practice problems

you're gonna get a light bulb of some kind

on how it works

[toxic aspen]

Yeah that's what i'm trying to do haha MATEMATIK is my first one :(

[regal harbor]

or just literally draw a diagram

try and reason thru it + watch youtube videos for further knowledge if you're really into the "why"

[toxic aspen]

2.5h in and i still dont get MATEMATIK

First year Uni

Yes.

[regal harbor]

she just doesn't seem to understand the scenario where the letters repeat

[toxic aspen]

^

[regal harbor]

MATEMATIK there are 3 letters that repeat 2 times

[toxic aspen]

The total amount of ways that MATEMATIKcan be arranged is 9! however i have to remove the amount of repeating ones, right?

MMATEATIK and ATEATIKMM. When i divide by 2! for the letter M does that take into account all the spots that the two MM can be in?

[regal harbor]

i don't really know the reasoning behind it, when i first learnt it i just took it as is

but that's my thinking process, yuo gotta divide by the duplicates

[regal harbor]

https://www.youtube.com/watch?v=1Uy2E2ncazg&ab_channel=EddieWoo this might help lol

3:25 he explains duplicates in the word

[toxic aspen]

Okay i think i'll just live with it and take it as it is. Gonna hvae to make a notepad before the exam so i dont forget it.

So for the word MISSISSIPPI n = 11! and then divide with the amount of dublicates? 4! for I, 4! for S, 2! for P?

$\frac{11!}{4!4!2!}$

[thorny flame]

Merineth

[toxic aspen]

Like so?

[regal harbor]

yea u gotta divide by the duplicates

that's how i have it memorized

[toxic aspen]

Oki i'll just memorize it then. Kind of hard for me since i generally remember stuff by knowing what they do

[regal harbor]

imo when you get to combinations it's more intuitive since you get to diagram it

[toxic aspen]

diagram?

[regal harbor]

what i mean is like

if the word problem has something like "4 places" you can draw it on a board

and reason thru it

[regal harbor]

its ok i forgot the quadratic formula like 7 times now

i forgot the amount of times i had to relearn it

[toxic aspen]

Who?

[toxic aspen]

Ooh neat i saw a lot of these examples when it came to arrangements of books?

[regal harbor]

yea, you can do the same for combinations

[toxic aspen]

But the arrangementes of books seem to be able to use the formula for permutations

[regal harbor]

is it a 100% that you'll have a word arrangment in the exams?

i only have combinations/permutation word problems for mine

i can send you an interesting combinations word problem you can solve

when you get to combinations

[toxic aspen]

I got plenty of examples to go through i think i'll be good but

How do i know when i "arrange" vs "select" to decide between combinations or permutations

for example :

In how many ways kan the letters MISSISSIPPI be arranged such that two S are not next to each other?

Is this permutations or combinations?

[regal harbor]

it's permutations since order matters

but you have to consider that "S" not being next to eachother

if the word problem asks: from a class of 15, in how many ways can you make a group of 4 or something like that

then its combinations

since order doesn't matter

https://www.youtube.com/watch?v=XJnIdRXUi7A&ab_channel=TheOrganicChemistryTutor 11:15 is more indepth on that

[toxic aspen]

Yeah i'm aware he covers that but i dont get it

I've watched like 5 separate videos about permutations and combinations

and it makes literally no sense

[regal harbor]

is it late for you rn

[toxic aspen]

no?

[regal harbor]

i asked because my brain's absolutely off when its past 9 PM

so what i'd suggest is

get pen/paper and draw it out

imagine these as spots that you can put stuff in

try n mess around

that's how i got a light bulb personally

https://www.youtube.com/watch?v=DROZVHObeko&ab_channel=KhanAcademy this explains it pretty well

it might not make sense, but if you practice enough it will

[toxic aspen]

I have 5 spots i can place the letters MIIIIPP

[regal harbor]

i think you don't even have to consider those 4 spots right here

just apply the formula, only the 5 spots matter

[median temple]

some things you memorise

some things you dont

i would memorise the formula behind perms and combs

but i would also know why

but for something like this it doesnt matter

or why 1+1=2

[toxic aspen]

like please stop typing

[regal harbor]

memorizing everything is a good way of not understanding anything

[toxic aspen]

RIght but the 5 _ spots are where i can place the letters. So isn't the question more of like "in how many ways can i arrange the letters MIIIIPP in these 5 spots?

[regal harbor]

yeah pretty much

[toxic aspen]

So if i permuate that

$\frac{7!}{4!2!}$

[thorny flame]

Merineth

[toxic aspen]

Shouldn't this be correct?

[regal harbor]

you shouldn't even care about the duplicates

the 4 spots are already taken care of

all you need is the 5 spots

MIIIIPP

you have 7 letters my bad lol

[toxic aspen]

I checked the book and it says $\frac{7!}{4!2!} \binom{8}{4}$

[thorny flame]

Merineth

[regal harbor]

im kind of confused on the 4! * 2!

[toxic aspen]

Yea i know!!! This is what i mean it's sooo confusing

You can't just memorize it, i have to understand it if i want to be able to solve other similar problems

[prime oxide]

its 4 x 3 x 2 x 2

[unreal kite]

they divide by 4!2! since you can permute the 4 I:s in 4! ways and the 2 P:s in 2! ways. You divide by the amount of permutations of each since all of them refer actually to the same arrangement.

[regal harbor]

its kind of the ALABAMA problem where you still have to consider the duplicates

[toxic aspen]

Why do we multiply with 8 over 4?

And is that combinations or permutations?

.close

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[sharp cobalt]

We have a>b>c sides of right triangle

And the angle C has 15 degress

And we want to find the areas for this possible right triangles

<@&286206848099549185>

[steep nebula]

maybe using the law of cosines? 🤔

[gilded plume]

Angle B would be equal to 75°

Hmm

[sharp cobalt]

I was thinking about that

Any ideas?

[topaz sinew]

@sharp cobalt Has your question been resolved?

[sharp cobalt]

<@&286206848099549185>

[topaz sinew]

@sharp cobalt Has your question been resolved?

[topaz sinew]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[minor compass]

Did they mean that the diameter is equal to the height? Bc the width is obviously way smaller than the semi circle here

[glass canyon]

it would be equal to the width, or probably would be better to call it the length, or the dimension that goes from left to right.

[minor compass]

That is not right though

You can see the semi circle is greater length then that

[minor compass]

I feel like I’m taking crazy pills here

https://tenor.com/view/mugatu-crazy-pills-gif-13247253

It’s not possible for the arch to be the same size as the width….

Can someone confirm this? Plz

[glass canyon]

are you saying that the arc length isn't equal to the width? if so, that would be correct?

[minor compass]

Yes

But they are saying it is

[glass canyon]

it's saying the diameter is equal to the width, which is correct

or 2r, for r being the radius

[minor compass]

Oh

Shit

So how do I find the arch length then?

[glass canyon]

probably just $\frac{\pi d}{2}$

[minor compass]

Area = pi r^2

[thorny flame]

fish

[glass canyon]

since it would be half the circumference

[minor compass]

Right cause circumference is c = pi D

[glass canyon]

yes

[minor compass]

Okok

So then it would be in this case 2w/2

So to set up the equation we do 2L + w + piw/2 = 30 ft

For the perimeter

[glass canyon]

are you using 2 in place of pi?

[minor compass]

And for area I. Did L*W + pir^2/2 = a

O sry

[glass canyon]

if so, yeah that's good

[minor compass]

Fixed that

[minor compass]

Ok

Did you take a look at the area equation too

[glass canyon]

yes

it's good

[minor compass]

Ok

[glass canyon]

do you need any help with the optimization part?

[minor compass]

Not yet I’m gona do it and prob mess something up then I will

[glass canyon]

alright

[minor compass]

I’m going to the bathroom so might take a minute

[minor compass]

Ok I might need some help now

[glass canyon]

alright

[minor compass]

Do I just put that into LW + pir^2 /2 ?

Or do I need to do something else first

[glass canyon]

i kinda forgot how to do this. let me solve it myself to remember

[minor compass]

Sounds good

[glass canyon]

ah, find the other dimensions, like plug in your value of w into the perimeter formula to find L, then you can solve the area (r is also just w/2)

[minor compass]

Yea

Are my calculations correct

Oh

[glass canyon]

i got 60/(4+pi) for w, though.

[minor compass]

How?

[glass canyon]

let me type my process

[minor compass]

I have to go to another building this is closing u can just send a picture of u want

[glass canyon]

i think your mistake is that l should be 15-w(1/2+pi/4)

but the process is right

[minor compass]

Which line is that

Where?

Oh

It should be the same answer though

Shouldn’t it be?

Ok I’ll try your way

[minor compass]

Can you send me a picture of your work

I agree with yo that wa my mistake

But I am having a hard time with the derivative

I got 15-pi*w for the derivative

I got the same answer still lol what am I doing wrong here

@glass canyon idk how but I still got the same answer

15/pi = w

Omggggg

I see why wow

Wait

I’m rly confused

<@&286206848099549185>

Oh I actually see why now

I am forgetting the pir^2 / 2 part

Ok I got w = 12

I rly hope that is correct

Um I am kind of getting it right but also wrong can someone help me plz

<@&286206848099549185>

[minor compass]

Hey are you there man?

[next kite]

wait what is ur issue ?

[minor compass]

I am trying to figure out what L is equal to

and then after that I have to maximize the area of window space

[minor compass]

I messed up L

[next kite]

wait

i keep getting confused on the way you wrote it

[minor compass]

Which part

[next kite]

first 3 lines

[minor compass]

Ok

[next kite]

const, obj and const?

[minor compass]

Ok the constraint

The objective

And the constraint in terms of one variable L

[next kite]

why does w/2 become 2w?

[minor compass]

I messed that part up

The L equation is what I’m trying to figure out

Wrong

[next kite]

ohhh i see

[minor compass]

He said it was 15 - w(1/2 - pi/4)

But I don’t think that is a good way of writing it cause I will need to derive it

I also got 15- w/2 - pi/4

Oh wait actually that part is finished

I need help with the next part

L = 15 - w/2 - pi/4

So I put that back into the objective equation and then derive

I’m having trouble with that part

Sorry I’m typing so much I’m really overtired and haven’t eaten anything

[next kite]

deriving or if youre supposed to put it back?

[next kite]

u should eat something

[minor compass]

First I need to put L into the objective function

[minor compass]

I kno

There’s no food I’m at school so I have 1 more after this then I can go eat

[next kite]

ngl im very bad at these types of problems

[minor compass]

Yeah these are annoying but could u just verify my algebra then?

[next kite]

so you're finding length and width?

[minor compass]

As you can see obj function is LW + pi r^2 / 2

Yea

Because it’s the area of the rectangle part plus the area of half of a circle

So LW + (pie r^2 )/2

So look

[next kite]

so the sum of the two areas?

[minor compass]

No

It’s an optimization problem

[next kite]

yes that

[minor compass]

It’s calc 1

[next kite]

yes

[minor compass]

Ok

Oh

I thought you meant something else

Yes

[next kite]

are u gonna use deriv to find max

[minor compass]

Yes

But first I need to input L into the objective eq

I am doing bad math right now though

I need help with the algebra part of it

[next kite]

which part

L?

[minor compass]

Mm the simplification of (L)w plus the. Other area

[next kite]

LW + pir^2/2

?

[minor compass]

Yea

[next kite]

you want to isolate L and W?

[minor compass]

Cause r = w/2 in this case

So it’s really annoying me

[minor compass]

Yea

Well I need to simplify the equation as much as possible before I derive it

[next kite]

lemme try

[minor compass]

I got this so far

[next kite]

woahhhh

wait

[next kite]

what is that for

[minor compass]

I plugged in L for the objective function

[next kite]

what did u get for L

[minor compass]

LW + (pi (w/2)^2 )/2

15-w/2-pi/4

[minor compass]

15-w/2-pi/4

[next kite]

hummm

i didnt get that

[minor compass]

What did you get

[next kite]

maybe i did something wrong

[minor compass]

Lmk if anyone else can help

[next kite]

yeah you are prob better off w/ other help

[minor compass]

Yea sry

[next kite]

im sorry

[minor compass]

Don’t worry you did your best

I’m gona ping helpers now tho

<@&286206848099549185>

I e been working on this over an hour

Shouldn’t rly take that long

[next kite]

what u did look right tho

[minor compass]

Yea the person helping me said he got the same thing but he disappeared

[next kite]

u just have to add the last 2 i think

[minor compass]

Well I got this

But the derivative part is kinda weird

[next kite]

whys that

[minor compass]

Idk it just didn’t add up

I can’t solve for w

Maybe idk what I’m doing

But it doesn’t make any sense

Like w can’t equal 0 I did something wrong

Oh wtf

It needs to equal 15

[next kite]

the last'

term

wait

[minor compass]

Ye

Ok

[next kite]

isnt it piw/4

[minor compass]

So I can move 15 over to other side then -15 = -w(1 +pi/2)

[minor compass]

Why

I factored the w out

Wait

Why over 4

Did I derive wrong

[next kite]

2pi*w/8

[minor compass]

Yea

Damn

What’s the shortcut for that

When you have a constant underneath

[next kite]

just take it out

if it makes it easier

like pi/8 *w^2

[minor compass]

Oghhh

[next kite]

idk if thats what u were asking

[minor compass]

Yea

And the constant stays the same tho

Cause u add it back in

[next kite]

i got 15-w-pi*w/4 for deriv

[minor compass]

Ok

[next kite]

can u double check for me

[minor compass]

Yeah sure

Yes

It is

So then when you set that to 0 how do u solve for w

[next kite]

keep 15 outside

i mean, on other side but what youre moving should be w so its positive i guess

[minor compass]

Eus

Yeaa

[next kite]

then factor w like u did

try that

[minor compass]

Ok

I know i can change the signs

But I’m not getting how to make it equal to 15 lol

Idk how to solve

Oh

Wait I do

[next kite]

u can keep the 15 positive

[next kite]

its like saying 15=4x, how would u solve that?

[minor compass]

I kno

15/4

I did -15 / (1+piw/4)

= w

Then I have to isolate the 15

So multiply the pi w / 4 back to other side

Idk if this is making any sense

Then bring four over

[next kite]

i think the 15 is positive

[minor compass]

So multiply it by 4

Ok

[next kite]

bc u move the -w-piw/4 to +w+piw/4

[next kite]

uhhhh

no need

[minor compass]

O yeah

Ignore that - sign in front of 15

[next kite]

u shld be good if u simplify the denominator under 15 no?

[minor compass]

How

Oh

[next kite]

it's just 1+pi/4

common denom

[minor compass]

No it’s not it’s pi w / 4

[next kite]

wait

why?

u factored w

[minor compass]

Oh

Damn

I’m slow

[next kite]

youre tired and hungry

[minor compass]

Right right right

[next kite]

[minor compass]

Haha

I need a new eraser after this

[next kite]

okay so fixing that what do u hav

[minor compass]

Um hint in

Hold on

[next kite]

there u go

[minor compass]

That’s it?

Rly?

Lemme check

[next kite]

dunno

u need to find max

[minor compass]

The obj now

Yea

I need to get L

Too

Oh wait

No

I have both now

[next kite]

yes

[minor compass]

So if I deriveitve again

[next kite]

u can use w to find L?

[minor compass]

I can see if it’s max

[next kite]

ye

[minor compass]

Yeah I need to do that

Wait lme do the derivative first

So that would be

Negative

So it’s max

Ok now

Put W in the contraint function???

W + piw/2 + 2L = 30

And if this doesn’t work I am giving up

[next kite]

what is your L

[minor compass]

I am going to solve for it

By plugging in W

Remember

[next kite]

yes

[minor compass]

I just said that aren’t u paying attention???!!

Ok sry

I’m doing it

Now

[next kite]

okokok

[minor compass]

Doe this look right so far

Cause this is cray

[next kite]

where did u get the const from

the uhh

[minor compass]

What 30 is given

[next kite]

L = 30-w-piw/2 ? or L = (30-2-piw/2) /2 ?

[minor compass]

L = 15-w/2-piw/4

What does that have to do with this

Bruh

[next kite]

bc ure using w to plug it in

and find L

[minor compass]

No we aren’t

That L in terms of W

We just got W

[next kite]

but u have w

and u said u were going to find L

[minor compass]

Yeah so now we plug in W to constraint

Oh

shit

Okok

Ah

Ahhhhhhhhh

Hehe

[next kite]

what did u find out

[minor compass]

That I don’t Wana do this anymore

[runic obsidian]

I can give a shot at it using TeX perchance

[minor compass]

Once I finish this i am done

Can’t handle if it’s wrong

Right now

Need food

Me hungry

Rghghhhhgh

Turning into dinosaur

[next kite]

ur L was L= (30-w-piw/2)/2, u found w=60/4+pi

[minor compass]

Wait wit ami doing info

Ong

Omg

[next kite]

why are u doing piw*4

[minor compass]

I was looking at the older sheet I haven’t changed it yet!

Omg

Ok

No wait

Wut r u talking abt

That is f’

[next kite]

ur old sheet

the 1st one u sent

had L= (30-w-piw/2)/2

[minor compass]

bruh

That is wrong

remember we did all this bc that was wrong

[next kite]

oh then u have a new L?

oh wait

[minor compass]

L is 30/2 - w/2 - piw/4

= 15 - w/2 - piw/4

[runic obsidian]

is w the width of the base

[next kite]

yeahhh okay

[minor compass]

And if that’s not right then I’m done with this for now!

[runic obsidian]

So your two parameters are of the rectangle.

The base width (w) and the height (h)

[minor compass]

Yes w is the base

And the half circle

[runic obsidian]

The semicircle at the top has the area $\frac{\pi w^2}{8}$ and perimeter $\frac{\pi w}{2}$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Ok

I think I have that

[runic obsidian]

The rectangle at the bottom has area $hw$ and perimeter $2h + w$ (excluding the top of it because it meets with the semicircle diameter)

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

so $A = hw + \frac{\pi w^2}{8}$ and $P = \frac{\pi w}{2} + 2h + w$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Yeah rats what I have

[runic obsidian]

P is fixed, i.e it's 30

[minor compass]

Wait

[runic obsidian]

so what you can do is write the height as a function of the width :3

[minor compass]

Oh ok nm yea I have that

[runic obsidian]

so if you have a given width, you can determine the height

$30 = w \left( \frac{\pi}{2} + 1 \right) + 2h$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Did that I got L = 15-w/2-piw/4

[runic obsidian]

For the sake of sanity

you can set (pi/2 + 1)/2 to be a constant k

so you don't need to keep writing it out

:3

[minor compass]

Aright

[runic obsidian]

$w = \frac{30 - 2h}{\frac{\pi}{2} + 1}$

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

Now we can plug w into the formula for A to get a formula for the Area given the height :3

But... You can be smart with it

[minor compass]

Where did you get that from

[runic obsidian]

30 -2h = w( pi/2 + 1)

divide both sides by (pi/2 + 1)

[minor compass]

So you’re solving for w instead of l?

We just solved for L

Y not out that in the area formula

[runic obsidian]

Actually yeah you can do that too

Two ways to skin a cat here

[minor compass]

Can we bc that’s how I did it

[runic obsidian]

Okie dokie :3

$h = \frac{30 - w\left(\frac{\pi}{2} + 1\right)}{2}$

frick

[minor compass]

It’s ok keep going

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Ok I got that

I simplified it

To 15-w/2-pi/4

So it’s easier to find the derivative

[runic obsidian]

Now we are trying to find the largest area. I'll ask you this: How do we find the largest area?

[minor compass]

We need to get the derivative

[runic obsidian]

Of what in respect to what?

[minor compass]

And find max

Derivative of w

[runic obsidian]

Derivative of Area in respect to w

[minor compass]

Oh makes sense

Aright

So is it cool if we use 15-w/2-pi/4

For f(w)

[runic obsidian]

$\frac{dA}{dw} = \frac{d}{dw}\left[ hw + \frac{\pi w^2}{8}\right]$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Huh

[runic obsidian]

Evaluate that while leaving h as a variable :3

[minor compass]

Why is h there we are deriving 15-w/2-piw/4 I thought

Oh

Wait

My bad

[runic obsidian]

Are we not trying to find the largest area?

[minor compass]

Jk

Kokomo

Typo

Ok

[runic obsidian]

You are onto something though :3

[minor compass]

Yea you input that for H

[runic obsidian]

now, lets quickly find $\frac{d}{dw} \left[hw\right]$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Ok

[runic obsidian]

$\frac{d}{dw} \left[hw\right] = \frac{dh}{dw}w + h \frac{dw}{dw} =\frac{dh}{dw}w + h $

cmon TeXit

$\frac{d}{dw} \left[hw\right] = \frac{dh}{dw}w + h \frac{dw}{dw} =\frac{dh}{dw}w + h$

TeXiT i BEG OF YOU

[minor compass]

Maybe no space at end

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

oh lol

thanku :3

[minor compass]

Of course

[runic obsidian]

Now we only need to put in the h once, and dh/dw once

[minor compass]

Ok

I’m trusting that part is correct cause it was quick

[runic obsidian]

it's chain rule

[minor compass]

Aright

[runic obsidian]

d/dx[ab] = a db/dx + da/dx b

[minor compass]

My school is dumb they didn’t teach us that way

But it’s fine

[runic obsidian]

oh, interesting

[minor compass]

I get it

Let’s just continue

[runic obsidian]

$ = \frac{dh}{dw}w + h + \frac{\piw}{4}$

[minor compass]

Have been on this problem for 2 hours

Want to go home

😦

[runic obsidian]

$= \frac{dh}{dw}w + h + \frac{\pi w}{4}$

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

now what's left is to plug in dh/dw and h

[minor compass]

Ok

[runic obsidian]

if you'd prefer, I can guide you through it later

or do it on paper and give you a copy with the rest to be done

up to this point

[minor compass]

Nah I needa do it now my ocd is abd

Bad

[runic obsidian]

i feel ya there

[minor compass]

I did everything you’re saying

[runic obsidian]

Okie

[minor compass]

But I just showed my work

[runic obsidian]

so what's left is to solve for w when dA/dw = 0

[minor compass]

So what did u get for the f(w)

[runic obsidian]

f(w) = h?

[minor compass]

Yes but I need to know if my work is correct

Yes

Like what is the actual equation lol

[runic obsidian]

here, of which the derivative is -(pi/2 + 1)/2

[minor compass]

That’s not f(w) because you first need to plug that into the A eq

That w in terms of L

[runic obsidian]

oh

A = f(w)

[minor compass]

Yeah what did u get for that

[runic obsidian]

oh i didn't do that or expand it

[minor compass]

Yes that is what I need help with ohm lll

Omggg lol

I need some one to check my work

We got L

Just plug it in if u Wana help

[runic obsidian]

$A = \frac{30 - w\left(\frac{\pi}{2} + 1\right)}{2}w + \frac{\pi w^2}{8}$

[thorny flame]

Arzela-Açaí Theorem

[minor compass]

Ur pretty smart btw

Yea I got that too but we can simplify it further

[runic obsidian]

oh it's a quadratic

[minor compass]

Yea

[runic obsidian]

vertex form

let me do that

let $\mu = \frac{\frac{\pi}{2} + 1}{2}$ so we can read it better and it stands out

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

$A = w \left(15 - \mu w + \frac{\pi w}{8}\right)$

[thorny flame]

Arzela-Açaí Theorem

[runic obsidian]

so we pulled out the w to the front

now

[minor compass]

Let me make sure that equals 15w-(w^2)/2 - piw^2/8

[runic obsidian]

i was trying to make it into quadratic form

[minor compass]

Yes I think it does

That’s fine

Makes sense

Continue

[runic obsidian]

15w - (pi/8 - mu)w^2

[minor compass]

Did u go to an Ivy League or something

[runic obsidian]

No quite the opposite currently

[minor compass]

Not in college?

[runic obsidian]

Anyway

Sorta community college for engineering lol

[minor compass]

Interesting how good u r at math tho

[runic obsidian]

I'm a calc tutor and I'm trying to get better at live-tex-ing

[minor compass]

Nice

[runic obsidian]

it's difficult over text and discord (sad)

[minor compass]

Ur fine at it

Anyway

Maybe they’re just slow

Ur fast

[runic obsidian]

15w - (pi/8 + 1/2)w^2

[minor compass]

Yes

[runic obsidian]

Now what we can do is let b = 15, and a = - (pi/8 + 1/2)

we have then

aw^2 + bw = 0

[minor compass]

So w =

[runic obsidian]

I'm completing the square here

[minor compass]

I got 60/4+ pi

Oh

[runic obsidian]

Also you can verify your answers

[minor compass]

Well I didn’t really do this step

I didn’t make it quadratic

[runic obsidian]

oh then you can solve for w and plug it back in

[minor compass]

I did it the really long boring annoying way my scjoool tight me

[runic obsidian]

so lets do that

[minor compass]

Yeah

Needa do that

Sry buuudd

[runic obsidian]

(aw + b)w = A

A' = (aw + b)' w + (aw + b)w'

[minor compass]

Nice

[runic obsidian]

A' = aw + (aw + b) = 2aw + b

[minor compass]

That makes much more sense

[runic obsidian]

now a = - (pi/8 + 1/2), b = 15

[minor compass]

Even tho I’ve never seen it like that before

[runic obsidian]

I tend to use variables to condense messy constants often involving pi

because writing it out at every step is annoying

[minor compass]

Bro how are y that fast at this lol

Yes this is much easier this way

[runic obsidian]

not writing it down lo

[minor compass]

True

[runic obsidian]

typing

[minor compass]

So what’s the final answer

I hope it’s the same I got

[runic obsidian]

A' = aw + (aw + b) = 2aw + b = 0

w = -b/2a = -15/(pi/8 + 1/2)

[minor compass]

Simplify

[runic obsidian]

120/(pi + 4)

[minor compass]

Damn I got 60

[runic obsidian]

Let me check the veracity using desmos rq

[minor compass]

Yep

Bro ur def an Ivy student

[minor compass]

I wonder where I went wrong

[runic obsidian]

i think i messed up somewhere

[minor compass]

Oh

Nm

U did?

I see

What does 60/4+pi look like

[runic obsidian]

I lost a half

[minor compass]

Alrig

Well

Now we can do the second part of the problem

[runic obsidian]

60/(pi + 4)

so you were right

just misread your format

[minor compass]

Which is plugging that into the constraint and getting L

[runic obsidian]

60/(pi + 4) is correct

[minor compass]

Or H

[runic obsidian]

getting A

[minor compass]

Ok but that’s just W

Yeah we stil need h and A

[runic obsidian]

i see

It says dimensions but doesn't really specify what dimensions. l and h are for the bottom rectangle

Anyhoo

[minor compass]

Yea but we have in objective eq the arc length

[runic obsidian]

Does it ask for the arc length

[minor compass]

It gives it

It asks for perimeter

[runic obsidian]

or just the box "around" the window

[minor compass]

Arc diameter = width

[runic obsidian]

yes

[minor compass]

Arc = c/2

[runic obsidian]

I'm asking what dimensions it's asking for you to put down

[minor compass]

It’s asking for LW because you can make the arc in w

Arc is just (pi(w/2)^2)/2

[runic obsidian]

oh okie yeah

then you plug it into l(w)

[minor compass]

Sry that’s the area

Yea

But first we need to plug it into the first eq

To get L

[runic obsidian]

L = l(w)

[minor compass]

So we needa plug in 60/4+pi into l(w) yes

[runic obsidian]

yes

but that seems like a symbolic trainwreck

[minor compass]

I know

That’s y I can’t finish this oml

Mental block

[runic obsidian]

does he want aforementioned symbolic trainwreck

[minor compass]

Yes

[runic obsidian]

or just a decimal answer

:horror:

[minor compass]

Good question

I think aforementioned train wreck is ok if it won’t take too long

So I can brush up on my algebra skills

[runic obsidian]

What you can do to check it is to plug it in verbatim into desmos

and see if it equals what desmos says it is

that's how I usually check veracity

[minor compass]

Ok

[runic obsidian]

if it's a symbolic trainwreck

[minor compass]

Yea I sometimes dip

Do

I just wanted to make sure the first part of this was right

I’ll finish this part

[runic obsidian]

okay, but yeah you did it correctly

[minor compass]

On my own

Yea

Thanks

A lot

[runic obsidian]

I don't like problems like this because I don't think the cost of effort just throwing in pi (which doesn't mesh nicely with rational numbers) in these problems is worth the skills students get from it

besides just patience?

It's the same idea but now it's just trying to not mess up where pi is

[minor compass]

The entire school I go to is terrible

They had us doing 100 derivative problems over and over till we burn out then test us on a trick question

[runic obsidian]

Uni calc is, from what I've seen, more lenient and has less of these types of gauntlet run problems

[minor compass]

Hws are harder than tears

Tests

[runic obsidian]

yeah

[minor compass]

Anyways I gotta run but it was nice chatting with ya, hope to see from you in the future.

[runic obsidian]

textbooks often do that

anyway see ya

[minor compass]

Bye

.close

[topaz sinew]

Channel closed

Closed by @minor compass

Use .reopen if this was a mistake.

[topaz sinew]

Available help channel!

Send your question here to claim the channel.

Remember:
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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[neon iron]

Please help me with this question.

[neon iron]

can someone help me pls

<@&286206848099549185>

<@&286206848099549185> please help

please help

<@&268886789983436800> why wont the helpers help lol theres no one

[bright trail]

Don't ping mods for this.

[neon iron]

sorry

its due tmr

[craggy haven]

also, don't immediately ping helpers after two minutes

[neon iron]

sorry i wont do it again

[bright trail]

The helpers are all just random volunteers.

[neon iron]

ok

thanks

[craggy haven]

people might help you, but frankly you've already gotten off to a pretty terrible start with your relationship with them

so good luck

[neon iron]

sorry its just that i really need to get it done

thanks for your help

[bright trail]

I would just find the area of each face of the object and then add all of the areas together.

[neon iron]

thanks 😄

ill try that

[topaz sinew]

@neon iron Has your question been resolved?

[topaz sinew]

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• Show your work, and if possible, explain where you are stuck.
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[fallow pivot]

Can anyone explain how to do this 🙏🙏

[junior nimbus]

whats the question

@fallow pivot

[fallow pivot]

Write each expression in radical form

I changed my answer though

[junior nimbus]

thats good

[fallow pivot]

Thanks

.close

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Closed by @fallow pivot

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[topaz sinew]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

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[ocean pewter]

I'm getting my answer in terms of parametric s and t, but I don't think that's correct

[ocean pewter]

I know that this is the definition of a projection:

Where the <a,b> is used to denote the inner product, which we are assuming is the Euclidian Inner Product (also known as dot product)

[dusk adder]

You notice that two vectors given for the basis of the span are not orthogonal

[ocean pewter]

Do they need to be orthogonal for a projection to work?

[dusk adder]

No, but you are double counting if project onto them separately and then add up

[ocean pewter]

I first defined S as

Then I plugged it into the projection definition above usingthe dot-product and I got

[dusk adder]

I'm not really sure where this is going

Did someone teach you a method like this

[ocean pewter]

I'm looking for the projection definition in my textbook right now

[dusk adder]

The definition for project onto a vector is this yeah

[dusk adder]

Where did this come from?

[ocean pewter]

I used t and s in the image as parameters to represent the span as a single vector

so I could then use the definition of a projection

Hello?

<@&286206848099549185>

[neon iron]

no

[topaz sinew]

@ocean pewter Has your question been resolved?

[ocean pewter]

<@&286206848099549185>

[urban grove]

Lemme take a look

[urban grove]

So, notice that this is the definition for projection onto a vector

What you want is projection onto a subspace

[ocean pewter]

So using the parametrics wasn't right?

[urban grove]

I don't think it'll get you anywhere

[ocean pewter]

Okay

[urban grove]

Have you learned about the Gram–Schmidt process yet

[ocean pewter]

Yeah, it's kinda annoying, but as long as I don't have to make it orthnormal by hand I'll be okay

I'm decent-ish with just the orthographic part

[urban grove]

Yeah that's one way you could try going about it, make the two vectors for the surface orthogonal

and then project v onto them

and then add together the two vectors you get

[ocean pewter]

Imma try that

[urban grove]

Another thing you could do is: calculate the normal vector to the surface, project v onto that, then subtract v minus the projection

That should also work

[ocean pewter]

what's a normal vector? Like a cross product?

[urban grove]

I think making the S vectors orthogonal and then projecting is more straightforward tho

[urban grove]

yeah

[ocean pewter]

Yeah, the textbook has that as the next unit, but I've heard about it before

[urban grove]

I think just go with the first thing I said then

[ocean pewter]

Yup, working on it

My new orthogonal set is:

<1, 1, 0> and <-0.5, 0.5, 1>

[ocean pewter]

Have I been doing it right so far? Cause if I am, I believe I just now have to add the two projections togethr

Giving me a final answer of <0.7, 4.3, 3.6>

[urban grove]

Lemme see

[urban grove]

how did you get the 2.5?

[ocean pewter]

5/2

[urban grove]

should be 6/2, no?

[ocean pewter]

I'm gettng 1*2 + 1*3 = 5

[urban grove]

v = [2, 4, 3] tho?

[ocean pewter]

Oh

[urban grove]

2 + 4 = 6

[ocean pewter]

I wrote down as <2,3,4>

[urban grove]

ah

[ocean pewter]

Let me redo some stuff 😬

Fixed it

[ocean pewter]

My new final answer is <1,5,4>

[urban grove]

seems good to me!

[ocean pewter]

Cool, I'm submitting it

It said it was wrong

Does it need to be orthonormal?

[ocean pewter]

☝️

[urban grove]

I don't think it being orthonormal would make a difference

Lemme think if something was wrong

oh wait I think I came up with an easier way to solve

[ocean pewter]

What is it?

[urban grove]

We know proj_S v is some linear combination of w1 and w2

so let's call it c1 w1 + c2 w2

[ocean pewter]

👍

[urban grove]

we also know that v - proj_S v is orthogonal to S

so that means (v - proj_S v) dot w1 = 0

[ocean pewter]

I kinda forgot that, but sure

[urban grove]

and (v - proj_S v) dot w2 = 0

so expanding that out,
(v - c1 w1 - c2 w2) dot w1 = 0
(v - c1 w1 - c2 w2) dot w2 = 0

now you have two equations and two unknowns (c1 and c2)

so you can solve for c1 and c2 and you'll be done

[ocean pewter]

But doesn't each system of linear equations split into 3 more because of how vector addition works

[urban grove]

No bc the dot product of two vectors is a scalar

So we're just saying scalar = 0, not vector = 0

[ocean pewter]

That makes sense

I'm having a hard time doing the dot products without error

[dusk adder]

Sorry, you could have tagged me directly with your reply. I didn't see it. Glad you found someone else to help with it though

[ocean pewter]

My bad, thanks for the initial guidance

I'm getting c1 = 1

amd c2 = 5/4

[ocean pewter]

Did you get the same solutions for the scalars?

My new new final answer is <1.5, -1.5, -1>

[urban grove]

Lemme check

[urban grove]

hmm I think I'm getting something different, can you show your work with simplifying out the dot product

[dusk adder]

Also cool trick when you have two vectors u and v and want to get orthogonal ones with the same span. If |u|=|v|, then u+v is orthogonal to u-v

[ocean pewter]

I did it on paper, but I got that v - c1 * w1 - c2 * w2 = <2 - 3*c1 + 2*c2, 3 - 2*c1 - 2*c2, 4 - 4*c2>

[dusk adder]

Typically faster and easier than Gram-schmidt for 2 vectors

[ocean pewter]

Then I did the dot-products to get the system

15 - 15*c1 = 0
18 + 2*c1 - 16*c2 = 0

[ocean pewter]

That's a good idea for this, let me try it

[dusk adder]

You should learn how to do Gram-schmidt though