help-41

[opaque maple]

lemme draw this rq lol

[frosty geode]

This no?

We need to find how much to cut from each corner

The length and width of the squares

[opaque maple]

ok don't mark length and width with x and y

that's what's messing you up

x is what we're cutting out of this

[frosty geode]

So what are the width and length?

Just w and L?

[opaque maple]

sure yeah

well maybe it's not even useful to name them

yeah actually save those letters for our box

[frosty geode]

How would you use the formula then🥲 the volume formula

[opaque maple]

• keep your picture, get rid of old x and y labels
• cut out squares from each corner and label their sides x

[frosty geode]

Done haha

[opaque maple]

great!!

see how that's like

[frosty geode]

Ohh I see and once we cut it it’s foldable

[opaque maple]

yes!!

[frosty geode]

What’s now?🥲

[opaque maple]

well now we can identify the dimensions of our folded box

let's start with the bottom

[frosty geode]

Wait we already folded it?

How do you know how much to cut tho to maximize the volume

I could’ve made the squares way bigger and get a tiny box

[opaque maple]

i should say:
now we can identify the dimensions that we might fold our box

since x is variable

[frosty geode]

Ohh oops

[opaque maple]

we can do a little calculus to maximise volume

the hard part is the visual and equation setup. it's ezpz after that promise haha

[frosty geode]

🥲🥲🤣

[opaque maple]

okok so the base of the box in your sketch

[frosty geode]

16 by 12?

[opaque maple]

that's the og cardboard

after we cut away those squares?

[frosty geode]

-4x?

[opaque maple]

think more in the lengths of the rectangle

so kinda yea

one side is 12

we take away 2x

then do the other side separately

[frosty geode]

Why do we need to separate if all squares are equal?

192-4x

[opaque maple]

you're really close but not quite

[frosty geode]

🥲

[opaque maple]

we can't just subtract all the x at once

[frosty geode]

Isn’t that logical tho?

We have an area and we want to cut 4 equal x’s out of it

So area minus 4x

[opaque maple]

hmm

so you're forgetting all the sides as well

you could do it like that but then you also need to remove the four sides

[frosty geode]

Aren’t the sides included in the area?

Ohh

[opaque maple]

remember we only want the base of the box

[frosty geode]

Cuz then they aren’t the bottom anymore

They get folded up

[opaque maple]

yep yep

so instead of subtracting all those at once

[frosty geode]

Damn🥲

[opaque maple]

we can do each side of the rectangle

[frosty geode]

So 16*6

-2x

Squared

[opaque maple]

where's the 6 from ?

[frosty geode]

Half of 12

One side of the rectangle

[opaque maple]

do one side at a time 😭

length is 16

and we reduce it by 2x

[frosty geode]

Ye one side is 16 by 6

[opaque maple]

OH

[frosty geode]

Wait wut

[opaque maple]

i see what you mean...

just describe l

in terms of 16 and x

[frosty geode]

Ohhh

You mean how much length would be reduced from L

So L-2x

[opaque maple]

yes! 🥳

[frosty geode]

Let’s gooo

16-2x

[opaque maple]

yusss

and similarly for the width?

[frosty geode]

And now we do it for each of the sides

[opaque maple]

yaya

[frosty geode]

(16-2x)^2 +(12-2x)^2

[opaque maple]

naurr

[frosty geode]

Whyyy🥲

[opaque maple]

we have the length and width of the rectangle

no need to square and add

A = lw

[frosty geode]

But we have 2width and 2 lengths

[opaque maple]

we only have one rectangle of the base

[frosty geode]

We cut length from both L’s

[opaque maple]

you correctly computed the sides of the rectangle

you just multiply them for the area

of the rectangular base

[frosty geode]

(16-2x)(12-2x)

?

[opaque maple]

yep!!

[frosty geode]

That doesn’t make sense to me🥲

[opaque maple]

can you update your drawing?

that might help

[frosty geode]

That’s as if we only took care of one side

What should I draw?

[opaque maple]

the base of the box

so the inside rectangle of the cuts

[frosty geode]

That’s what it looks like to me based on this logic

Cuz we took care of only one length and one width

Basically we cut everything that’s shaded in

And are left with another width and length to cut

Cuz it wasn’t squared

[opaque maple]

ok a lot to clarify here

[frosty geode]

🥲

My brain😫

[opaque maple]

ok i made a terrible sketch

i think firstly you want to think of x as the side-length of the squares

we aren't subtracting squares or areas right now

just measuring the lengths of the inner rectangle (the base of our box)

[frosty geode]

OHHHH

That’s the bottom only

[opaque maple]

sorry i know this is so hard over text haha

yes we want the bottom!

[frosty geode]

This is already a better explanation than what my professor did lmao

[opaque maple]

okok well from here the base is good!

so we only need the height

[frosty geode]

If we only subtract the corners how does 16-2x and 12-2x not include the sides

[opaque maple]

hmm maybe more concretely if we say x = 3
then inside our 16 * 12 rectangle we'll have a 10 * 6 rectangle

but if you want to do them all separately

big rectangle - 2 rectangles - 2 rectangles - 4 squares

16*12 - 2*3*10 - 2*3*6 - 4*3*3

[frosty geode]

This converted to (16-2x)(12-2x) is missing the 4 rectangles then🥲

OH WAIT

[opaque maple]

that's good news 😭

[frosty geode]

The rectangles are -32x and -24x???

[opaque maple]

woah what?

[frosty geode]

16(-2x) and 12(-2x)

[opaque maple]

nonono

these are subtracted

[frosty geode]

Then I’m lost again🥲

[opaque maple]

the length was sixteen and we shortened it by 2x

similarly for twelve

[frosty geode]

But if you expand this term

[opaque maple]

foil!

[frosty geode]

You’d get 192-32x-24x+4x^2

[frosty geode]

Wut?

[frosty geode]

Which is what you said here

[opaque maple]

that's how i learned to expand sorry

[opaque maple]

this is great

[frosty geode]

Big rectangle -2 rectangles -2 rectangles -4 squares right?

[opaque maple]

however you want to think of it yes

[frosty geode]

Ye so that’s what I meant these are the 2 rectangles

[opaque maple]

i find it easier to just think of the sides of the inner rectangle but it's the same!

[frosty geode]

And 2 other rectangles

I finally have a logic to grasp on then🥲🥳

[opaque maple]

16(-2x) is def not the same as (16 - 2x)

[frosty geode]

Well ye but you could also expand the factors

[opaque maple]

these are not two rectangles

these are the sides of a single rectangle

[frosty geode]

Wouldn’t that be the folded up sides?

[opaque maple]

take a look at this again

are you okay with the inner rectangle being the base of the box?

[frosty geode]

Yes but I disagree that it’s only -2x

X is the corner right?

2 x are two corners

[opaque maple]

x is the length of the squares

[frosty geode]

Ohhh

X times X is the corner

[opaque maple]

yes!

[frosty geode]

But x times x is x squared🥲

[opaque maple]

if you want the area of the squares yes

[frosty geode]

So why is it not 16-x^2

[opaque maple]

16 is a length

not an area

[frosty geode]

How can we substract area from a length

[opaque maple]

we can't!

[frosty geode]

I really need to get this practically to make sense of it🥲

[opaque maple]

that's why it's not 16 - x^2

[frosty geode]

Ohhh

So 2 x is the length of the square twice?

[opaque maple]

yes!

[frosty geode]

OHH WE ONLY SUBTRACT THE LINE ITSELF

[opaque maple]

YUSSS

[frosty geode]

Shorten the line

From both sides

[opaque maple]

🥳

you got it

[frosty geode]

Finally😭

[opaque maple]

😭

[frosty geode]

Hahaha I’m so slow

[opaque maple]

no worries lmaooo

[frosty geode]

And then we shorten the width too

[opaque maple]

yuss

[frosty geode]

Okay good😭😭

[opaque maple]

breakthrough

alrightyyyyy

[frosty geode]

Eureka

[opaque maple]

V = lwh and now we only need the height

[frosty geode]

Oh god

[opaque maple]

it's easy promise

imagine folding it up

[frosty geode]

I got nothing 🤣

[opaque maple]

back to this again

[frosty geode]

How do we even know how far is the center from the edges

[opaque maple]

we have it algebraically with x

[frosty geode]

To know how much of them is being folded up

[opaque maple]

that's the point of making it variable. so we can optimize for volume

we basically want a formula that will give us the volume for these boxes depending on how big a square we cut out

and that visual from before should match our sketches and you can see what part becomes the height when we fold it up

[frosty geode]

Height=(16-2x)(12-2x) with something added

Idk a square a root

Dividing it by smth idk🥲

All 4 sides become the height

[opaque maple]

they kind of prop up yea

[frosty geode]

But we don’t know the 4 sides

[opaque maple]

we do!

we know how high they'll be when we fold it up

[frosty geode]

-32x and -24x?

[frosty geode]

How so?

[opaque maple]

look at this again

and compare to the values on our sketch

[frosty geode]

ABC forming a triangle?

Idk what to look for

[opaque maple]

we cut away those squares

and then they kind of go away in 3d when we fold it up, right?

[frosty geode]

Yea

But idk how to convert this to numbers

No idea what I’m supposed to look at

[opaque maple]

we don't have numbers

we're using variables

[frosty geode]

So how do we know the height

[opaque maple]

specifically x

[frosty geode]

-4x?

Idk😭

X is gone

We cut it off

Ohhh

=0??

[opaque maple]

x is the lengths of the squares remember

nono

don't label these squares as x

we're measuring lines on this box

and some of them are length x

[frosty geode]

I’m about to cry🤣

[opaque maple]

it's okay lol

you're def overthinking this

[frosty geode]

We have one side 16-2x

Now we need to fold it up

What does it do to the x

Yea no idea

[opaque maple]

so looking at this. we've established the base of the box, so don't worry about 16-2x and 12-2x anymore

the corners are physically cut

and the four side rectangles are lifted up

how high is each side rectangle?

[frosty geode]

Am I supposed to know the height formula for this

[opaque maple]

no we have it written down in the picture 😭

[frosty geode]

Wait really😭😭

[opaque maple]

yes haha

[frosty geode]

I only know area😭

[opaque maple]

ok ima give this one to you cause i think i'm not helping at this point

the height will be x

[frosty geode]

WHAT

I’d have never guessed it

[opaque maple]

try to fold that box and see that haha

[frosty geode]

OH MY GOD CUZ THE CORNERS ARE X

SO THATS THE LENGTH

FOLDED UP

[opaque maple]

yeppp!! hahah

[frosty geode]

Bruhhhh😭

[opaque maple]

i'm so sorry 😭

[frosty geode]

Hahahah I have overthought this

[opaque maple]

ok but that's legit the hardest part

[frosty geode]

Thank you😭 you are making it make sense in my brain

Step by step

[opaque maple]

your brain has expanded with visualizations needed for calc

[frosty geode]

Yes definitely ❤️

[opaque maple]

alright now we can write down V

[frosty geode]

So basically volume=(16-2x)(12-2x)(x)

[opaque maple]

yep!

[frosty geode]

Holy shit

[opaque maple]

noiceeee

so the whole point of that was to express the volume in x

now we can use calc to optimize the value of the volume wrt x

[frosty geode]

Why does derivative work here

To find the maximum value?

[opaque maple]

yep!

[frosty geode]

So I need to use product rule?

Or just expand this

[opaque maple]

def expanding is easiest

[frosty geode]

Calculating lol

[opaque maple]

same haha

[frosty geode]

Is this correct

Do I at least know algebra 😭

192x^

Idk why I wrote 196

[opaque maple]

looks good!

[frosty geode]

🥳

I love algebra

[opaque maple]

home sweet home

[frosty geode]

Now derivative of this?

[opaque maple]

yep!

[frosty geode]

Wait how does taking derivative here works

Cuz we need to find the maximum

But this will give us the minimum

Since the leading term is positive

It would be a local minimum

[opaque maple]

well finding the derivative will help us calculate any extrema

so mins and maxes

[frosty geode]

Oh I see, I guess I’ll wait and see🤣

So this should not be equal to 0 right?

Cuz that would give us minimum

[opaque maple]

we do want to set it equal to zero

and find the x for which that's true

[frosty geode]

Ohh cuz we get 2 x’s

[opaque maple]

those will be critical points to let us know where the mins and maxes are

typically we would include the bounds here as well

[frosty geode]

What if it was just 12x-112 for example then the problem would just not make sense,

?

[opaque maple]

correct, we want to tie things to reality here. there may be extraneous solutions

but at this point there really shouldnt be any lol

[frosty geode]

I see, I’ll use the quadratic formula rq

I knew we would get an ugly root

16 sqrt13

Tf do I do with that

[opaque maple]

hmm wait can you show your work

that's not quite right

[frosty geode]

(-112)^2 is 12544

[opaque maple]

ahh the 24 divides both

both terms

[frosty geode]

Hm?

Oh

Well still what do I do with it?🤣

[opaque maple]

right we prob don't need an exact answer for this lol

i think you can enter the derivative to see what they want for the next section

[frosty geode]

I hate calculus🥲

Wait they don’t even want the volume😭

[opaque maple]

ok good we can round haha

[frosty geode]

🤣🤣

How do we even find x then

[opaque maple]

so when you set the derivative to zero, that gave us a critical point

we want to see the behaviour near there

do you call it the first derivative test?

[frosty geode]

The behavior near 112+- 16 sqrt13 over 24?!🫠

[frosty geode]

Second I think

[opaque maple]

yeah but let's just round to like 5 decimal places for now

[frosty geode]

I did that before

Can we just do the second derivative test I find it a lot easier

Than finding nearby points

You just have to see if it’s decreasing or increasing right?

[opaque maple]

sure!

however you want

[frosty geode]

Yupp

So 12x-112

Ffs

X=28/3

Well that’s bigger than 0 at least right

So it means that this is a local minimum iirc

Meaning it’s increasing

I think

[opaque maple]

well you found a critical point for the second derivative test

which will tell you concavity

[frosty geode]

It tells you if the slope is increasing or decreasing right?

So if it’s increasing it should tell you how the curve will behave at that point

So it’s increasing

Isn’t that the behavior

[opaque maple]

this might help
https://www.desmos.com/calculator/lz0dkdx0oi

[frosty geode]

Honestly whenever they ask me to analyze it by graph I’m just lost🤣

But from what I understand here it’s increasing

[opaque maple]

the value until your critical point is negative

so the slope is decreasing

[frosty geode]

Wait wut

[frosty geode]

This is positive

[opaque maple]

that's the value of the critical point for your second derivative test

[frosty geode]

Then I forgot how to test it

[opaque maple]

oh hold on

yeah that's the wrong value

[frosty geode]

Isn’t it 12x-112

12x=112

/12

X=28/3

[opaque maple]

f(x) = 192 - 56x^2 + 4x^3
f'(x) = 192 - 112x + 12x^2
f''(x) = -112 + 24x

[frosty geode]

Fuck I forgot to multiply

[opaque maple]

nw haha

[frosty geode]

So x=14/3

[opaque maple]

took me forever to see where my work differed from yours 😭

[frosty geode]

I’m sorry🥲🥲

[opaque maple]

nwnw

and yea 14/3

you can see that on the desmos

[frosty geode]

Well that’s still positive tho

So how is the slope negative

[opaque maple]

yes that value being positive is not important

[frosty geode]

Then I have no idea what’s happening 🥲

[opaque maple]

it tells you where concavity changes

[frosty geode]

What are we trying to do with this

How does it tell us dimensions?

[opaque maple]

yea we're getting lost in the sauce

[frosty geode]

🤣🤣

[opaque maple]

start over from the derivative haha

we want a max value

[frosty geode]

How does derivative tells us dimensions

I thought it’s used for the volume

[opaque maple]

it let's us choose an x

which we're optimizing volume for

[frosty geode]

But we don’t need the volume

[opaque maple]

yes but we have all our dimensions in terms of x

get our x that's nice and we have all three sides

[frosty geode]

So it’s a stupid question that set us on a path that doesn’t help us solve the final step?

[opaque maple]

no we have to find this x with the first derivative test

[frosty geode]

How do we know what x is supposed to be

[opaque maple]

i guess we did a tangent with 2nd test mb

so to find a local max we find the critical points

that was our nasty sqrt13 thing

[frosty geode]

I always used to find local maximum and minimum with the 2nd derivative test

[opaque maple]

so maybe we just have different names haha

the idea is we want to see where the original function changes from positive slope to negative slope

[frosty geode]

Yup!

[opaque maple]

which is why we test values between the critical points

great!

[frosty geode]

Oh

I never plugged in values

[opaque maple]

right

[frosty geode]

I just saw if it’s bigger or smaller than 0

[opaque maple]

exactly

so remember we want to put bounds on this too

[frosty geode]

Well this is bigger than 0

[opaque maple]

that's not what we're comparing to zero 😭

also that was the wrong test tangent from before

mb

[frosty geode]

I’ll just follow you idk what’s happening

[opaque maple]

it's ok

i'm all over the place too sorry lmao

[frosty geode]

No it’s a stupid question

[opaque maple]

from here!

[frosty geode]

And I have one with a square after this😭😭

Do I plug it into the first or second derivative

[opaque maple]

we want to look at the slope

the first

since we want to know when the og was increasing and then decreasing

[frosty geode]

So 12x^2-112x+192

[opaque maple]

yes!

[frosty geode]

And I just choose a random x?

[opaque maple]

nooo

we solved for that x already

[frosty geode]

So I plug in the catastrophic answer from before

[opaque maple]

2 actually

yes but let's round so we don't kms

[frosty geode]

😭🤣

[opaque maple]

they want 3 digits, we'll play it safe with 5 for now haha

[frosty geode]

7.07036 and 2.262965

[opaque maple]

great!

[frosty geode]

I plug both to the derivative?

[opaque maple]

well you'll just get zero

we want the behaviour around those

[frosty geode]

Wtf😭

Derivative+1?

[opaque maple]

look at this again rq
https://www.desmos.com/calculator/e4irocety8

[frosty geode]

Second derivative?

[opaque maple]

we don't need 2nd

so in the graph the green is the first deriv and you got the zeros

[frosty geode]

What am I looking for when looking at graphs

Ohh yea the 0’s

X intercepts

[opaque maple]

ideally we'd bound all our crit points

[frosty geode]

What is bounding

[opaque maple]

upper and lower bounds

x can't be smaller than zero or greater than six

[frosty geode]

Also how tf does that graph transfer into a box

[opaque maple]

physically

[frosty geode]

Why 6?

[opaque maple]

these functions are just describing the values of volume

oh 2x is twelve and we can't cut longer than our box length

[frosty geode]

That’s so abstract

Ohh

[opaque maple]

yep haha

[frosty geode]

Why did you put 0 and 12

[opaque maple]

oh i had it on 6 but it cut it off lol

you should ignore that 😭

[frosty geode]

Oh xdd

[opaque maple]

ok so we only want to look at crit points like:
0 --- 2.26297 --- 6

we can check the sign of the derivative between them

to confirm the slope is pos/neg

[frosty geode]

Positive?

Cuz it’s above the x axis

[opaque maple]

yep!

visually that's easy

[frosty geode]

And then negative

[opaque maple]

but if you don't have a graph, this is where you'd plug it in

[frosty geode]

How do you do it by algebra

[opaque maple]

yeah

[frosty geode]

Plug in where?🥲

[opaque maple]

f'(1) and f'(5) would be between those crit points

[frosty geode]

Ohhh

Plugging nearby numbers

[opaque maple]

and they'd be pos then neg

yep!

[frosty geode]

Lemme do both

[opaque maple]

sure!

[frosty geode]

Wait

Don’t we have just one answer

Cuz 7 is impossible

So the only answer is 2.etc

[opaque maple]

actually yeah lmao!

[frosty geode]

No way I found smth by myself😭

[opaque maple]

it would be good in general to check in case we messed up and we can confirm it's an actual local max

but let's move on there's no need haha nj

[frosty geode]

I see that’s smart ye

[opaque maple]

sooooo actually we're basically done

we have the dimensions of the box in terms of x. just plug em in

[frosty geode]

Wait I got 92

Is that the height then?

Oh wait no

[opaque maple]

nono x = 2.26297

[frosty geode]

That’s a positive answer

Means that the slope is decreasing

Righhht?

[opaque maple]

ummmmm

[frosty geode]

I calculated v’(1)

[opaque maple]

look here again https://www.desmos.com/calculator/ugbfb1llus

oh oh oh

i see what you're saying

"slope is decreasing" can only be found w 2nd derivative test which we're not doing right now

[frosty geode]

The line I mean

[opaque maple]

the slope being positive means the original function is increasing

[frosty geode]

From 0 to 2.etc

[opaque maple]

yep!

[frosty geode]

Ohhh

And then that would mean that until that point it’s increasing and when we reach that point where it starts decreasing that’s the maximum

[opaque maple]

so that line (our theoretical volume values) maxes out there

yep!!

[frosty geode]

How would we find that tho we would have to calculate so many points

Cuz even if we did v’(3) we wouldn’t get the extra answer

[opaque maple]

you computed when it was exactly zero

[frosty geode]

Ohh right!

[opaque maple]

when you set the derivative to zero

yaya

[frosty geode]

Sp we just saw what happens before that

Cuz after the 0 it has to change direction

[opaque maple]

right just a heuristic if we don't want to graph this

[frosty geode]

We I just had to know if it’s a minimum or a maximum

[opaque maple]

exactly

[frosty geode]

Omg I get it😭

[opaque maple]

yusss 🥳

alright well now we have the height x

[frosty geode]

So x is the height

[opaque maple]

and we know length and width were 16 - 2x and 12 - 2x

[frosty geode]

And now we can find the other things by plugging in x?

[opaque maple]

exactly

[frosty geode]

Is it simply 16-2(2.etc)

And 12-etc

Separately ?

[opaque maple]

yep!

[frosty geode]

😭😭😭

Thank you so much🥲

You need to replace my professor

[opaque maple]

so proud of you !! 😭

you really stuck through it

ima eat now hahaha

[frosty geode]

Enjoy your food!!

[opaque maple]

if you have this much tenacity you'll do fine in math no prob!

yeah gn!!

[frosty geode]

My brain has melted😭

[amber wasp]

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[frosty geode]

I’m stuck, how do I finish this?

[frosty geode]

I put the derivative=0?

Now what?

[solar gust]

You can simplify a little

[frosty geode]

How so?

[solar gust]

-pix/2 = -2pix/4

[frosty geode]

I don’t get it

I move it to the other side of the equation,

[solar gust]

No

[frosty geode]

What does that mean then

[solar gust]

Just rewrite

[frosty geode]

How does that simplify it I’m confused

How do I find what x is equal to

[solar gust]

Wait the thing above is your original function or your derivative ?

[frosty geode]

Derivative

[frosty geode]

These are the steps

That’s step 5

The derivative

I equaled it to 0

To find what x is equal to

[solar gust]

Ok

I see

[solar gust]

Its equal to -x -2pix/4 +pix/4 = -345/2

So -x -pix/4 = -345/2

[frosty geode]

That’s so ugly, I thought it has to be x=

[solar gust]

Wait

[frosty geode]

Damn

[solar gust]

factor this by x now

[frosty geode]

x(-pi/4)?

Idk

[solar gust]

Not quite

[frosty geode]

No idea

Idk how we even got it over 4

Or what happened to all the other terms

[solar gust]

I simplified fraction

[frosty geode]

How do I do it?

[solar gust]

By adding them ?

Since we have common denominator

Its totally legal

[frosty geode]

Wait you can add fractions with different denominators?

Oh wait 2?

[solar gust]

Theyre not different

[solar gust]

Here

Theyre not

[frosty geode]

How did you change it to 4

[solar gust]

Multiply 2 on top and bot

[frosty geode]

Ohhh

Hold on lemme do that

Like this?

[solar gust]

Yes

And now

[frosty geode]

Nicee

[solar gust]

Factor by x

[frosty geode]

x(-1-pi/4)

[solar gust]

Wunderbar

[frosty geode]

Wut

[solar gust]

And then you can divide now

[frosty geode]

What’s wunderbar

[solar gust]

Wonderful

[frosty geode]

Ohh lmao

[solar gust]

In german

[frosty geode]

Thank you

[solar gust]

You're welcome

[frosty geode]

So I divide by the factor?

[solar gust]

Yes to have x = ...

[frosty geode]

How do I simplify the catastrophe on the right lol

Multiply the fractions and flip?

[solar gust]

No

Wait

Its not (-1-pi)/4

Its -1 -pi/4

[frosty geode]

Oh yea yea! Whoops😅

[solar gust]

So it would need you to write it as (-4-pi)/4 before flipping

And i think you should be fine

And if not ask wolfram

By saying : ,w calcul

,w 2-3

[frosty geode]

Like this no?

[grizzled pagoda]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by yakubros

[frosty geode]

Oh

[solar gust]

No

[solar gust]

See

[frosty geode]

Why do I need them both to be negative

[solar gust]

?

[frosty geode]

I don’t get it

Tbh

It was -1-pi over 4

[solar gust]

Yes

Which is the same as (-4-pi)/4

And i can flip this legally

When -1 -pi/4 isn't flipping legally

[frosty geode]

Why not?

Where did the -1 go

[solar gust]

I multiply by 4 on top and bot to have one fraction instead of two fractions

Like we did before

[frosty geode]

Ohh cuz it’s the same as -1/1

?

[solar gust]

As an advice, you should refresh some basic fractions manipulations

[frosty geode]

Yea I’m weak with complex fractions🥲

[solar gust]

Thats really dommage that you are stuck with this and doing some derivatives questions that are technically harder

[frosty geode]

Wait you get this and then you combine them? -4-pi over 4?

[frosty geode]

Hahah some areas are weaker for me like fractions🤣

[solar gust]

No get back to the previous line

[frosty geode]

I meant to send this^

I multiplied -1 by 4 and 1 by 4

So -4/4

[solar gust]

Yes

Now combine

And put back into the frac

[frosty geode]

So

[solar gust]

No

[frosty geode]

How🥲

Oh -pi

[solar gust]

Yes

So you have that -345/2 / (-4-pi)/4

So flip it and multiply it

Gives

[frosty geode]

So x=1380pi over 8?

[solar gust]

,w simplify -345/2 / (-4-pi)/4

[frosty geode]

Wth

I got this

[grizzled pagoda]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by yakubros

[solar gust]

You didn't flip

[frosty geode]

Fuck

Then divide the -1380 by -8?

..

[solar gust]

,w -345/2 * 4/(-4-pi)

[grizzled pagoda]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by yakubros

[frosty geode]

Where did 4 over -4-pi come from

[solar gust]

You can cancel the 2 with one of the 2 from 4

Cuz 4 = 2x2

[frosty geode]

Wut

I thought we already got past 4

[solar gust]

Well

Wait

[frosty geode]

I’m here

[solar gust]

$\frac{a}{b}\cdot \frac{c}{d} = \frac{a}{d} \cdot \frac{c}{b}$

[grizzled pagoda]

YakuBros

[solar gust]

You can swap denominator when product

[frosty geode]

What step are you mentioning

[solar gust]

This

[frosty geode]

But I don’t have 4 anymore

[solar gust]

Idk whats this

[frosty geode]

I got this right

I multiplied -345 times 4

And 2 times (-4-pi)

[solar gust]

You can simplify by 2 top and bot

[frosty geode]

Ohh like divide the whole thing by 2?

[solar gust]

Top and bot yeah

And get same result as wolfram

[solar gust]

Here

I gtg

Have a Nice day

[frosty geode]

Thank you

.close

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[regal swift]

Any ideas where I messed up?

(Stopped cause my work doesn’t even look close to the answer)

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@regal swift Has your question been resolved?

[red river]

Hello 👋🏽 , I see the first line has some mistakes

I think you forget the term +6y from original eq

also distribute properly -5 in -5y'

L{ -5y' } = -5 L{y'}

[regal swift]

Ohh ure right

Idk how i forgot a whole term lmao

[red river]

= -5 s L{y} + 5 f(0)

[red river]

haha dw

also be careful when factoring a common term

I think you factor s^2 L{y} - 5s L{y} into (s^2 - 5) L{y}

you forgot the s

[regal swift]

Oh right

I messed it up really badly wow

I’ll give it another go thanks 😅

[red river]

Ok np 👍🏽

[regal swift]

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[prime apex]

i'm assuming that this is probably an easy question in terms of the mathematics involved, but i don't understand a view things

[prime apex]

all i can do is graph both equations, i can only graph y = px-5 only knowing that it is a tangent and intercepts the y axis at 5

i feel like i just need to konw where the tangent intercepts the parabola and then it would seem pretty easy

[rough trail]

use the tangent formula, y = f'(a)(x - a) + f(a) and equate it to the given tangent to solve

You can also find where the tangent intersects the parabola by equating the tangent snd parabola

intersection of f(x) and g(x) can be found via f(x) = g(x)

[prime apex]

ahh yes ofcourse simultaneous equations thank you

ill try that

and the interception has to be x>0 as well okay

i understand it now

nvm i don't understand it

i just get stuck at (x+3)(x-1) = px-5

[rough trail]

you are trying to find where they intersect, yeah?

[prime apex]

yeah

i cannot find it

[rough trail]

Yeah it is pretty nasty but I'm checking to see if the tangent formula thing might work a little better

[prime apex]

okay thanks

i've factorised it incorrectly...

i mean does that even matter its just nasty

[rough trail]

No it doesn't matter. Are you sure this question works? I did a slider on Desmos and I don't think there is a real solution. It doesn't have a y intercepts at 5 for any values of p. nvm it is at -5 not 5

[prime apex]

it works

if i set g(x) = f(x)

can i then factorise the equation to x^2 +x(2-p) + 1 = 0

and then use the discriminate? where the discriminate must equal 1?

[tulip tapir]

discriminant

[prime apex]

right

[rough trail]

well it must equal zero if it's a tangent

[prime apex]

oh yeah 0

that would work?

[rough trail]

it wouldn't give you the solution, it would give you number of solutions

and you still have two variables, p and x

[prime apex]

wait im just gonna try it to make sure though

it gave me p=0 and p=4

but p > 0

so p=4 which indeed works

i mean then i don't even need to know where they intersect and i know p

[rough trail]

yes that actually makes sense

[prime apex]

it works

i tried it on another equation and it actually works

idk why it just works because i got p=10 and it works

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[split sail]

Can I get some help with Wolfram

[split sail]

I need to input this value

[split sail]

More clear ig

.close

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[fierce edge]

stupid question but how do we know that M'X is collinear to XC

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@fierce edge Has your question been resolved?

[zinc sapphire]

If you're finding the shortest distance for M to D, and then the shortest distance from D to C, and then reflect M, you're still getting the shortest distance, which is a line

The second you modify any of the values, the way the lines move will change because it's all about the SHORTEST distance (which will be a line)

Suppose M'X were not colinear with XC, then it wouldn't be the shortest distance, yet MD+DC would still be, seeing as M'X=MX=MD, so it would be absurd to say otherwise

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[south flare]

I need help with the 3 one

3 one 3 subdivision

[covert onyx]

Now, what do you see?

[amber wasp]

@south flare Has your question been resolved?

[split sail]

yup! remember the properties of iscoceles triangles to solve this one

[south flare]

No

It hasn’t been resolved

I also need help with the 2nd subdivision

[hollow cape]

What are you still hung up on?

[south flare]

I am trying to solve the 2nd subdivision but couldn’t do it

I was just reading the rules or the isoceles triangle

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[keen pawn]

I have to write these in the if-then form, would like my answers checked

[keen pawn]

1)If a matrix's determinant isn't 0, then it is invertible
\
2)If a function is differentiable, then it is conitnuous
\
3) If a function is integrable, then it is continuous
\
4) if a function is a a polynomial, it is rational

[vast spade]

looks good

[compact yarrow]

Objection

You've written 2 and 3 in the same manner, yet one says sufficient and one says necessary

Surely this should trigger an alarm

[keen pawn]

wait

If a function is continuous , then it is integrable

[compact yarrow]

Now we're talking

[neat wind]

🔥 🔥 🔥

[keen pawn]

Thanls

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[solemn escarp]

prove that if a,b are integers, a>0 and there exists only one integer pair q,r such that
b = aq +r where 2a < r <3a

[solemn escarp]

i dont really get how to do this

the other solutions in the book were confusing

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[fresh mantle]

So basically

Half of the numbers must be 1

[glad pulsar]

oh yeah

[fresh mantle]

Do u understand that part

[fresh mantle]

Mhm

[glad pulsar]

what next

[fresh mantle]

So 5 can be any number from 0 to 9

[hoary rover]

(there are no two digit prime numbers that are the product of two one digit numbers)

[fresh mantle]

So 10⁵ possibilities

[sterile nymph]

No?

[fresh mantle]

?

[glad pulsar]

no

[sterile nymph]

0 and 1 and 4 and 6 and 8 and 9 are not prime

[fresh mantle]

Oops

Sry mb

[sterile nymph]

So you can choose between only 2, 3, 5, 7

[glad pulsar]

what

why

[fresh mantle]

Yea yea

[fresh mantle]

Cuz if u choose

[glad pulsar]

oh

[fresh mantle]

4

[glad pulsar]

yea

understood

[fresh mantle]

Then 4*1 also

Yep

So basically 4⁵ possibilities

And

[sterile nymph]

And then there are odd 1s vs even 1s

[fresh mantle]

Yea

So 2*4⁵

So 2¹¹

[glad pulsar]

why 2*?

why 2

[fresh mantle]

Place one in 1st place

[glad pulsar]

ok

[fresh mantle]

1st 3rd 5th 7th 9th

Or

[glad pulsar]

ohh

[fresh mantle]

2nd 4th 6th 8th 10th

2 cases

So *2

[glad pulsar]

wow it was that easy

[fresh mantle]

Ye

[glad pulsar]

Thanks have great day

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[fresh mantle]

Anytime! You too have a great day

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[empty yew]

Can anyone spot my mistake

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@empty yew Has your question been resolved?

[untold shoal]

maybe your integral of e^x*sinh(x)

also brackets

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[acoustic lodge]

(∀ε > 0)(∃n0 ∈ N)(∀n ∈ N)(n ≥ n0 ⇒ |an − a| < ε).

[acoustic lodge]

so how do i read this

[fresh mantle]

Lmao I was about to ask that-

I think it is

[lilac osprey]

Like the symbols?

[fresh mantle]

(For all epsilon greater than zero) (there exists n belonging to natural numbers)

For all n belonging to natural numbers

U mean that?

[lilac osprey]

*n0

[acoustic lodge]

yes

[fresh mantle]

Yea

[frank zealot]

For all epsilon is strictly greater than 0, there exists n0 a natural number, for all n is a natural number, for all n is a natural number, n being greater or equal to n0 implies that |an - a| is strictly less than epsilon

good luck with this one

[acoustic lodge]

and /an-a/ is distance around what

[fresh mantle]

Yep so
Reverse A is "for all"
That e thing is "epsilon"
Reverse E is "there exists"
That e in 2nd bracket "is belonging to"
That arrow is "implies"

[acoustic lodge]

i know ahhh

[fresh mantle]

Oo

Ok

[acoustic lodge]

but the meaning is for

like convergetion i think right

[outer hull]

this seems to be the e-d definition for sequences convergence

[acoustic lodge]

convergetion hm

okey

[acoustic lodge]

thanks

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[outer hull]

i would say this definition is a bit confusing for me tho

this was the definition in hammack

if you have studied e-d definition for limits before, this should be fairly familiar

[acoustic lodge]

techically L is number that sequence is around

and epsilon is just arounf that

L

[acoustic lodge]

is L accumulation point

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[sand dagger]

how do they assume that ${V}_{n} \subseteq O$
couldnt it also be the other way? like the element of the base which has x is larger than the neighbourhood

[grizzled pagoda]

reg

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@sand dagger Has your question been resolved?

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@sand dagger Has your question been resolved?

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[sweet sentinel]

If {V_n} is a basis, then for all x in X, for each nbhd U of x, there exists V in {V_n} such that x in V \subseteq U

by the defn of a basis for a topological space

[empty yew]

https://tenor.com/view/jagten-the-hunt-the-hunt2012-mads-mads-mikkelsen-gif-24314799

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@sand dagger Has your question been resolved?

[split kraken]

what Sara said is correct. if $B = {V_n}_{n \in \N}$ is a base, and $O$ is any nbhd of $x$, there exists a $V_n \in B$ such that $x \in V_n \subseteq O$.

this is by the definition of a topology generated by a basis

[grizzled pagoda]

higher!

[split kraken]

{V_n} is not one set. it is an entire basis

the book is just using (what I consider) unclear notation tbh

[amber wasp]

@sand dagger Has your question been resolved?

[sand dagger]

ahh okay i thought Vn have to cover only the whole space

got it ty @sweet sentinel @split kraken

[sand dagger]

[sand dagger]

wait miku

[sand dagger]

can you pls share this server

[split kraken]

it's a private server, so I cannot, sorry

feel free to add it to your server though

also, if you're done here, you may close this channel

[sand dagger]

fair

.close

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[rocky fjord]

my teacher got y - bar = 20/7

i got y - bar 2.5

let me show what i did

i dont see where i went wrong, i just used dy instead of dx

<@&286206848099549185>

(ping me if u can help)