help-28

[glossy valve]

Combustion

[main ginkgo]

lim of sin(x)/x as x->0 is 1

solved

[torn jolt]

Sicko mode

How do you type that fast

[main ginkgo]

i don't lmao

i type on desmos

then copy and paste

[torn jolt]

How?

[main ginkgo]

ctrl + c, ctrl + v

[torn jolt]

Thas crazy

[quartz verge]

This step was acc really smart

Nice job dude

[main ginkgo]

[brittle steeple]

wtf cracked

@full marsh

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[little fractal]

Hello, good afternoon, I need help with this combinatorics exercise please. I don't know how to approach the exercises.
Ex.20) The elevator of a building carries 10 passengers and can stop at any of the 12 floors of the building.
a) Distinguishing between people:

1. In how many ways if on the 10th floor exactly 3 people descend?
2. In how many ways if at most one passenger descends on each floor?
   b) If no distinction were made between people, what would be the answers to the questions posed in a?

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[neon fulcrum]

Have you tried anything?

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[jade igloo]

I need this to go into standard form

sorry wrong one

[dull seal]

So what is standard form?

[jade igloo]

standard form not from

[dull seal]

Typo

[jade igloo]

yeah

[dull seal]

The question stands

[jade igloo]

yeah I'm just confused on the completing the square part

[dull seal]

do you have any try?

[jade igloo]

yeah I got down to the x side equals: x^2-6x+9, and the y side equals: y^2+30+225

I just don't know how to simplify it down further

[dull seal]

So you have x^2-6x+9

What is that equal to?

()^2

[jade igloo]

(x-3)^2 ?

[dull seal]

Yeah

From 222 we take out 9

That would be…

213 right?

[jade igloo]

wouldn't it be -222-9 because we move the 222 to the right side

[dull seal]

I didnt move anything yet

[jade igloo]

oh okay

[dull seal]

The equation would be

(x-3)^2 etc=0

So

How many “y” u have?

[jade igloo]

what do you mean like (y+15)^2 ?

[dull seal]

Yeah

15^2 is 225

Until 213 is 12

So that -12

(x-3)^2 + (y+15)^2 -12 =0

[jade igloo]

yeah

then move the 12 right

that makes sense

[dull seal]

The question was

To write in standard form

So this part u already know

👍🏻

[jade igloo]

rad

thanks

👍

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[muted ruin]

Okay so I know one and two

but don't know how to solve 3&4

[ember shadow]

3 is basically the same as 2 (same principle at least)

so is 4

[muted ruin]

I don't know what that means

so I divide?

[willow patio]

7 over x+10

[ember shadow]

no, you solve it like other exponential problems

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[wide zodiac]

What is the parallel and perpendicular equation of y=1/2x-6

[glacial pasture]

you can find infinite such lines

go ahead and get some

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[shadow mango]

Hi, so honestly i don't know how to solve this question, could someone help me with this one?

The y coordinate of a point is 2, find the x coordinate of the point if the distance of the point a(1,3) is the root of 16

[unique linden]

let point be (x,2)

now use distance formula

ull get a quadratic

solve it

[shadow mango]

i don't get it, i mean. why is (x,2)? besides. the coordinate of the point is the root of 16

[unique linden]

umm no

root of 16 is 4

meaning

distance of (x,2) from (1,3) is 4

[shadow mango]

oh, and it is 2 because the y coordinate of it is 2

once i solve the formula i'll get a quadratic?

[unique linden]

yes bro

[shadow mango]

tysm

[unique linden]

yes yes

now close d damn channel

[shadow mango]

i know i have to, i just want to see if i can solve it first since i'm insecure about my answers

is that okay?

or do you want me to close it anyways?

[unique linden]

well fine

leave it open sis

[shadow mango]

got it

thank you @unique linden

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[barren jolt]

Where is the mistake here?

[neat aspen]

"or"

you don't necessarily conclude both simultaneously

[barren jolt]

But when we solve quadratic equations, we say that both the solutions are correct

[grand hatch]

±any number ≥0 is impossible (other than any number=0)

[willow patio]

"or" states that if any of the statement is correct than the argument is true

My english was not enough there oof

[barren jolt]

I did not quite get your point

[neat aspen]

only one of x - y >= 0, y - x >= 0 need to be true

not simultaneously both

the mistake is assuming both are simultaneously true

[willow patio]

Assume he have 2 statements p and q and
r ≡ p V q

[barren jolt]

Yes I thought of your point, but as I said earlier, when we take x² = 25, both x=5 and x=-5 are correct simultaneously, right?

[hot herald]

x=5 OR x=-5
will satisfy the equation
x can't simultaneously hold two different values

also the +- in that first step is inappropriate

[grand hatch]

The square root function only returns nonnegative values

[barren jolt]

As the degree is 2, I have taken +-, is this wrong?

[hot herald]

is wrong, hence why I said it was inappropriate

that's not where the +- comes from

[barren jolt]

Then where does it come from in case of quadratic equations?

[grand hatch]

It comes from x^2-a^2=(x-a)(x+a)

If you have x^2=a^2, you write (x-a)(x+a)=0, so x=a OR x=-a

x=a or x=-a doesnt comes from applying sqrt to both sides and saying x=±a because √a^2=±a

[hot herald]

using $x^2 = 25$ as the example, taking the square root of both sides
$$\sqrt{x^2} =\sqrt{25}$$
$$\underbrace{|x|}_{\text{not } x} = 5$$
and casework / definition of absolute value from there leads to $x = \pm 5$

[glossy valve]

ℝαμΩℕωⅤ

[barren jolt]

But as the degree is 2 in case of my question, shouldn't it have 2 solutions as I have written?

[hot herald]

there'll be 2.5 cases

the point is applying square roots will give
$$\sqrt{(x-y)^2} \ge 0$$
there is no $\pm$ generated here

[glossy valve]

ℝαμΩℕωⅤ

[hot herald]

from the square root property, that leads to
$$|x-y| \ge 0$$

[glossy valve]

ℝαμΩℕωⅤ

[hot herald]

and THAT leads to \
$x - y \ge 0$\
OR $-(x -y) > 0 \to x - y < 0$

[glossy valve]

ℝαμΩℕωⅤ

[barren jolt]

So, my mistake was in assuming both the conclusions simultaneously, right?

[hot herald]

and technically there's more stuff behind that as well when doing casework, esp for more complex examples

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@barren jolt Has your question been resolved?

[barren jolt]

Here you have said that x-y>=0 or -(x-y) > 0
So, x-y>=0 or y-x>0
So, can't I say x = y?

[hot herald]

you can

[barren jolt]

But then it comes back to the same thing

[hot herald]

i attached the equality case to the greater than case for simplicity

you can keep that separate if you wish

[barren jolt]

So what's the ultimate conclusion to my question?

Which step was wrong and why?

[hot herald]

the first step

like i said

[barren jolt]

The +- one?

[hot herald]

and the conflation of OR and AND

[barren jolt]

Ok. Thank you all for your help.

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[brittle parcel]

how do I find this angle?

[brittle parcel]

nvm

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[torn jolt]

Could anyone help out with this? Ty

[vivid jungle]

Your approach?

If I could forget everything and started again, I wouldn't want someone else telling me the solution for it

Its one of those problems

😓

[torn jolt]

I'm confused to the point where I have no clue where to begin 😭😭

[vivid jungle]

Do you know the sum of roots?

[torn jolt]

Yeah -b/a

[vivid jungle]

So how much in this case?

If alpha + beta + gamma = S, then alpha + beta = S - gamma

^^ This is HINT 1

[torn jolt]

Okay so alpha+ beta would be -6 - gamma here then

[vivid jungle]

S is 6 not -6 here

[torn jolt]

Oh yep right -b

[vivid jungle]

So now you can replace the denominators

beta + gamma - 4 = 2 - alpha

[torn jolt]

Okay yeah

[vivid jungle]

Can you proceed from here?

[torn jolt]

The denominators replaced

[torn jolt]

Let me give it a shot

[vivid jungle]

Yup

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@torn jolt Has your question been resolved?

[torn jolt]

Nope couldn't proceed

[vivid jungle]

Were you able to simplify the expression?

[torn jolt]

I don't think so im pretty much still stuck at the exact same place (forgive my stupidity)

[vivid jungle]

No issues

So beta + gamma - 4 = 2 - alpha

And similarly for the other 2

So the first term in the expression can be written as $(2-\alpha)^3$, do you see this?

[torn jolt]

Yep did that bit

[glossy valve]

fukwerint

[torn jolt]

Ohhh yeah

[vivid jungle]

So you now need to find $(2-\alpha)^3 + (2-\beta)^3 + (2-\gamma)^3$

[glossy valve]

fukwerint

[torn jolt]

Mhm

Do I try the identity?

[vivid jungle]

Now put $y = (2-x)$ in the original equation and simplify

[glossy valve]

fukwerint

[vivid jungle]

Tell me what you get

[torn jolt]

I don't get this bit

[vivid jungle]

So if you put y = (2-x)

And simplify the equation

You would get a new equation

If $\alpha$ is the root of the original equation, now $2-\alpha$ would be the root of the new equation

[glossy valve]

fukwerint

[vivid jungle]

Do you get it?

Because you put y = 2 - x, so when x = alpha, y = 2 - alpha

[torn jolt]

Original equation refers to the f(x)=x^3... Bit right?

[vivid jungle]

Yup

See, you know that $\alpha$ satisfies $x^3 - 6x^2 + 5x + 1 = 0$

[glossy valve]

fukwerint

[torn jolt]

Yeah

[vivid jungle]

Now suppose you put $y = 2 - x$, and you simplify

[glossy valve]

fukwerint

[vivid jungle]

And you get some polynomial in y, say $g(y) = 0$

[glossy valve]

fukwerint

[torn jolt]

Where's y tho

[vivid jungle]

You created $y$, namely, $y = 2-x$

[glossy valve]

fukwerint

[torn jolt]

Ah okay

[vivid jungle]

Do you understand this transformation?

You can ask if any issues

[torn jolt]

I understand y=2-x I'm struggling to understand what a new polynomial with p(y) would look like

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[torn jolt]

.reopen

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✅

[torn jolt]

Ohh wait I think I got it

[vivid jungle]

You need to expand the new polynomial

It would look like a cubic polynomial with different coefficients

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[cunning bronze]

this is statistics - Hypothesis testing with T statistics where n<30 and std deviation of population is unknown. When you are finding out whether or not to discard the H_0 hypothesis you are supposed to use T statistics here, like finding out the critical value by using T table, my question is: if you are given a confidence level to test for discarding H_0, and its a one tailed test for the right side (i think this is irrelevant but idk) Are you not supposed to divide the alpha by 2 in this case? since its T distribution.

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[cunning bronze]

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[brittle parcel]

in solving this triple integral, why is z between -arctan and arctan

[brittle parcel]

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[worthy topaz]

I am stuck on the spearman coefficeinet question

[worthy topaz]

After ranking all the the numbers from biggest to smallest i got

[fiery pasture]

isnt that just a 🔌question

[worthy topaz]

yeah the answer was supposed to be like

0.9554 and i keep getting 1.04

i was so confused and i though my ranking was wrong

[fiery pasture]

let me look at your working out

[worthy topaz]

i re did it and now i got 0.919

[coral salmon]

You shouldn't have a rank of 5 there

Because the 4.5 covers 4 and 5 so your next rank should be 6

[worthy topaz]

owh a stupid mistake

thanks for helping out

[coral salmon]

It's okie lol let me know if you get the right answer

[coral salmon]

Nwsss

[worthy topaz]

yeah i got it

[worthy topaz]

thanks for that

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[vestal verge]

How do I do question 7

[vestal verge]

I set up the integral $$\int^b_1f(x)dx = \sqrt{b^2-1}-\sqrt{2}$$

[glossy valve]

Nathan

[vestal verge]

That’s all I have so far

[civic bay]

by the FTC, if the antiderivative of f(x) is F(x), then this integral becomes F(b) - F(1)

[vestal verge]

I wrote that too but I’m stuck there aswell

[civic bay]

which looks a lot like the difference on the right

well look at the F(b) term

[vestal verge]

Yeah but I can’t quite figure out F(x) looking at it

[civic bay]

see that it exactly matches with √(b² -1)

[vestal verge]

I thought $$\sqrt{x^2-1}$$

[glossy valve]

Nathan

[vestal verge]

But that won’t work for the other one

[civic bay]

so if F(b) = √(b² - 1), then F(x) = √(x² - 1) by straightforward substitution

oh wait you’re right

[vestal verge]

Yeah

[civic bay]

oh

look closer at the provlem

[vestal verge]

That was my roadblock

[civic bay]

it’s √(b² + 1)

not -

[vestal verge]

Oh I wrote it wrong on my whiteboard

Oops

[civic bay]

so F(x) = √(x² + 1)

[vestal verge]

Okay so it is

Yeah

[civic bay]

which you can differentiate to find f

[vestal verge]

It’s

$$\frac{x}{\sqrt{x^2+1}}$$

[civic bay]

yes

[glossy valve]

Nathan

[vestal verge]

That?

Okay

[civic bay]

yeah

[vestal verge]

Alright thanks

[civic bay]

happy to help

[vestal verge]

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[glossy valve]

kisnar

[full forum]

Please don't occupy multiple help channels.

[craggy tapir]

3 integrals, of what?

[hot herald]

separate how exactly

[craggy tapir]

if you are referring to 1/x, 1/2sqrtx and 1/2, no its wrong

[hot herald]

no, you can't split fractions like that

[craggy tapir]

yes, its wrong

[civic bay]

if you're actually trying to solve this, the sqrtx is a little annoying so maybe try letting u = sqrtx

[low echo]

This is question ?

[civic bay]

then the integrand becomes 1/(u^2 + 2u + 2) and the differential becomes:
u = sqrtx
du = 1/(2sqrtx) dx
dx = 2sqrtx du = 2u du

so your integral becomes the integral of 2u/(u^2 + 2u + 2) du

[low echo]

Hey see

Its

[grand hatch]

If you ever see sqrt(x) in an integral, consider the substitution u=sqrt(x), because sqrt(x) is contained in its own derivative

[low echo]

Basically 1/((√x+1)² +1)

[civic bay]

now this is something you can probably evaluate, doesn't look like the denominator factors nicely for you to use partial fractions but the denominator also looks like (u + 1)^2 + 1

which might be useful

you factored u^2 + 2u + 2?

[low echo]

Hm

[civic bay]

oh you mean you did the second integral using partial fractions

yeah that makes sense

i'm talking about this first intgeral after you do the u sub

[glossy valve]

kisnar

[civic bay]

you get this

i'll save you the trouble but you won't really be able to nicely factor it

unless you feel like doing partial fractions with complex roots

but i do notice that 2u / (u^2 + 2u +2) = 2u / ((u+1)^2 + 1)

[nimble gyro]

This one is very easy

[civic bay]

and if you let w = u + 1, then u = w - 1, then you get 2(w-1)/(w^2 + 1) dw as your integral

[civic bay]

when you split this into two fractions by separating the terms in the numerator, the first integral can be done with another substitution and the second integral can be solved using arctan

[low echo]

The factor theorem states that a polynomial of any degree can be reduced to quadratics and linear factors with real coefficients

No i mean partial fractions work for any degree polynomial

[civic bay]

when you're writing the form of the partial fractions, you separate based on the factors of the denominator, the numerator is always some arbitrary polynomial that's one degree less than that factor

[low echo]

its actually not , coz you can generalise it

[civic bay]

so if i have an irreducible quadratic factor, the numerator will be an arbitrary linear term

you also build up the power in the denominator for repeated roots

that's pretty much the main rules

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[dense turret]

[full forum]

@dense turret Has your question been resolved?

[torn jolt]

The answer is C. Range is the same when the angles of projection are complementary. There is a formula to calculate range. Try to prove it yourself. Also, range is the greatest when angle of projection is 45 degrees if I remember correctly.

Also the speed must be the same when the projectile is thrown

That is already there in the question so yeah

[final juniper]

Range = v^2 sin(2θ)/g
Time = 2vsin(θ)/g

where v is speed at which the object is thrown

and θ is the angle

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[frozen coral]

This is not something you should post here ig

[low echo]

Bro is much forward

I appreciate !

[frozen coral]

Bro copied the prompt from ChatGPT as well

[sacred yarrow]

Imma choose China

[rare dock]

what the fuck is going on in here

[neat aspen]

curry rice goes hard

[clever sluice]

India not having briyani is a crime

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@torn jolt Has your question been resolved?

[neat aspen]

i would have thought tikka masala or butter chicken would be the india stereotype

[clever sluice]

I mean kichidi is alright ig

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[hollow helm]

If two players are playing scrabble, and they are drawing their 7 tiles each from the bag of 100 tiles. As long as the tiles remain unknown to both players, than they will always both have equal odds to pull any given tile..................in other words, wether my opponent selects his 7 tiles, followed by me selecting my 7 tiles, OR.....my opponent selecting 1 tile, followed by me selecting 1 tile, until we each select 7 tiles each.....no matter which of these "draw" methods you chose, both players have equal chance to draw any 1 specific tile..............Even tho in the first example, the second player would be drawing from a pool of 93 tiles opposed to the initial drawer, drawing from a pool of 100 tiles, because the draws themselves remained unknown until the conclusion of the whole process, the probability for drawing a specific tile remains the same for both players..........my question is simply......would Bayesian inference suggest the same thing I just said? Or would Bayesian inference suggest if I draw second, after my opponent has drawn his 7 tiles, my odds cannot be the same as his, because even tho we don't know which specific tiles he has, we know the number, and that by definition changes my odds.

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@hollow helm Has your question been resolved?

[hollow helm]

No it's 5:54 in the morning and I'm asking dumb questions sbout Bayesian inference related to scrabble

Who would possible be up at this hour to answer such a dumb question

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[pseudo cape]

#linear-algebra message

[rare dock]

hi frosst

how goes learning probability?

oh oops

[pseudo cape]

[polar valve]

whats your question?

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[wintry hatch]

is an eigenspace just the set of all eigenvectors for a given matrix?

[wintry hatch]

this is what my lecture notes says but im tryna wrap my head round this ahaha

[vivid jungle]

Eigenspace is the set of all vectors that satisfy the equation

Its a vector space

Suppose it has 2 vectors, v1 and v2

So you know $Tv_1 = \lambda v_1$ and $Tv_2 = \lambda v_2$

[wintry hatch]

right

[glossy valve]

fukwerint

[vivid jungle]

So you know that $T(av_1 + bv_2) = \lambda(av_1 + bv_2)$ also

[glossy valve]

fukwerint

[wintry hatch]

im just confused because I thought that each eigenvalue has a eigenvector to go with it, is the eigenspace not just the set of all eigenvectors?

[vivid jungle]

For any a and b

Each eigenvalue may have multiple eigenvectors

So if the set you are talking about contains $v_1$ and $v_2$, it would also contain all vectors of the form $av_1 + bv_2$

[glossy valve]

fukwerint

[vivid jungle]

Tell me if you have any issues

[wintry hatch]

hmmm i see, im just trying to look at a worked example we did in class also as well as reading this

[vivid jungle]

Whats the confusion you have

[wintry hatch]

oh wait i think I understand now, so if I have a eigenvalue = 2, then the eigenspace is all the vectors that are Tv = 2v or is this wrong?

[vivid jungle]

Exactly

Correct

[wintry hatch]

ah right got ya

[vivid jungle]

Eigenspace "associated with that eigenvalue"

[wintry hatch]

I was looking at this example and had no idea what was going on till now ahahaha

can an eigenspace has more than 1 thing in it? so you see here we have (z,0,z). could it have another vector in there too?

[vivid jungle]

(z,0,z) means infinite vectors

(1,0,1), (2.1,0,2.1), (2.5,0,2.5), ...all the vectors are in there

[wintry hatch]

ah okok, can i just ask as well what does the span thing mean?

i am unsure on this too, trying to study and learn new content while revision for exams at the same time is tough. not really grasped this content as well as i wouldve liked

[vivid jungle]

Span is defined for a set of vectors

Suppose you have a set of vectors (it may have finite or infinite vectors)

Then the span(that set of vectors) is another set of vectors, which contains all the *linear combinations" of those vectors

By linear combination of vectors, I mean:
2v1 + 5v2 + 7v3 is a linear combination of v1,v2,v3

[wintry hatch]

so for this case here if span = {(1,0,1)}, we say that (1,0,1) is the span because we have the eigenspace as (z,0,z) and so we can write allthe eigenspace vectors as (1 * z, 0, 1 * z)?

i feel like ive worded this awfully i apologise

[vivid jungle]

span is not an independent thing

span is an operation on a set of vectors

[wintry hatch]

so say like if the eigenspace was (2z, 0, z) would the span be (2, 0, 1) ?

[vivid jungle]

Its like span(a set of vectors) = another set of vectors

So suppose span({1,0,1}) would be (the set of) all vectors of the form k * (1,0,1) where k is a real number

[wintry hatch]

so in this example here would you say k = z to get all the vectors in the eigenspace?

ohhhh wait i think iam understanding now actually

[vivid jungle]

z is nothing but a representation for any real number

And so is k

[ashen crescent]

An easy example would be to see that the span of the canonical 3d basis vectors is all of R^3

Because any vector in R^3 can be described as a linear combination of the basis vectors {(1,0,0), (0,1,0), (0,0,1)}

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[subtle prairie]

Suppose that a company's production Q, as a function of the amounts of capital K invested and labor L employed, is given by the Cobb-Douglas production function

Q: RxR→R: (K, L)-> K^2/3 L^1/3

What is the minimum amount that the company must spend on capital and labor to achieve output q if a unit of capital costs k euros and a unit of labor costs l euros? This amount will of course depend on the numbers q, k and , which we assume are strictly positive

[subtle prairie]

So this is the problem I was doing

It was going pretty well but I don’t know how to solve the equations at the end of the page

How do I find the critical point L,K ,lambda

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[pulsar mortar]

Need help with question i

[low echo]

Question 2?

Three cylinders like this

Max efficiency !

Now we need to know radii of circle for proceeding

Also idk if three cylinders have equal diameter !

@pulsar mortar hii

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@pulsar mortar Has your question been resolved?

[pulsar mortar]

Oh sorry I was afk

I’ll get you question 2 now

The three circles have equal diameters

The height of the cylinder is 19.95 I think

When your explaining could you do it in the simplest terms

My teacher hasn’t taught us any of this so I’m really lost on what to do

[low echo]

Ok so

Circle diameter is 6.65 cm

We need to find the side of this equilateral triangle

This is a upper view of our setup

[pulsar mortar]

So you’ve made it into a 2d shape ?

By just showing one of the sides

[low echo]

We re just viewing it from top to make work easy

[low echo]

Yeah !

[pulsar mortar]

Ok

[low echo]

Ok now wait one min

Ok then

See this

@pulsar mortar /

[pulsar mortar]

Yes

[low echo]

We can get the value of x using trigo

Total length of a side will be 2x + 2r

So can you calculate?

[pulsar mortar]

no

[low echo]

See

tan 60° = x/r

Or x = r tan60°

r = 6.65

,w (tan60°)×6.65

[glossy valve]

Results provided by WolframAlpha

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

Requested by itzmecairo

[low echo]

You see length of one side is approx 11.52

[pulsar mortar]

Ya

[low echo]

Area of equilateral triangle give by

√3/4 a²

So area = √3/4 (√3×6.65)²

Area × Height will yeild the volume

,w √3/4*(√3×6.65)²

[glossy valve]

Results provided by WolframAlpha

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

Requested by itzmecairo

[low echo]

Approx area = 57.45

,calc 6.65*3

[glossy valve]

Result:

19.95

[low echo]

Now volume = 57.45×19.95

,calc 19.95*57.45

[glossy valve]

Result:

1146.1275

[low echo]

So its 1146.1275 (approx) cm³

surface area = 2×triangle area + 3×height × side length

,calc 257.45+319.95*11.52

[glossy valve]

Result:

804.372

[low echo]

So approx surface area = 804.372 sq cm

[pulsar mortar]

I gotta go sleep but could you just explain what question 3 i is asking me to do ?

[low echo]

Its asking you to calculate the total surface area of the prism

That i have already explained and calculated

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[torn jolt]

Have I started of this integration by substitution question correctly ?

[grave elm]

whats that 2sincos(1/2x)

[low echo]

Bro

[torn jolt]

WAIT

THAT COS IS NOT MEANT TO BE THERE

[grave elm]

then you still need to apply chain rule

[low echo]

Why didnt you split it

[torn jolt]

Ignore the cos that’s there by mistake lol

[low echo]

Into two different integrations ?

no substitution required

3/sin²x/2 = 3 cosec²x/2

[torn jolt]

We haven’t learned that yet, my teacher has just told us to do substitution

[low echo]

Integration of cosec²x/2 is -2cotx/2

[low echo]

what

You can atleast use your logic of algebra

[torn jolt]

How so ?

[low echo]

Else i dont see a easy way to solve using substitution

a+b/c = a/c + b/c

[low echo]

^

[torn jolt]

Oh so rewrite as respective trig identities and integrate those parts ?

[low echo]

Yeah

[torn jolt]

That makes a lot more sense, thanks for your help

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[torn jolt]

.close

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[slow rover]

Hello, I have pretty complicated mathematical problem to solve. Here are the details:

There is a machine that generates 5000 units at a frequency of 57.8(703) mHz.
Every 35000 cycles, 4% of current number of units is taken away and stored separately in a cache.
Provide this cache value after 2^28 seconds.

So far I solved it for 2^21 seconds. The result is 8178,24 units.

The idea is to figure out that 5000 units are generated every day. Since generation can be considered as a event or cycle in the theory of frequency, it means that 1 unit = 1 cycle.
Based on that it's known that every 7 days 4% of current sum of units is stored in a cache.
After first 7 days we have 1400 units stored (C) and 33600 left (G) (C = 5000 * 7 * 0.04 = 1400, G = 7 * 5000 - C = 33600)
After 14 days we have 1400+2744 units stored (C = (G + 5000 * 7) * 0.04 = 2744, G = G + 7 * 5000 - C = 65856)
After 21 days we have 1400+2744+4034.24 units stored (C = (G + 5000 * 7) * 0.04 = 4034.24, G = G + 7 * 5000 - C = 96821.76)
After 2^21 seconds nothing more is stored in a cache because it's before day 28 so the result is 1400 + 2744 + 4034.24 = 8178.24 units

2^28 seconds means that I pretty much can't do it this way and I have to create a formula.
Please help.

[outer marlin]

do you know the answer

[slow rover]

no

[outer marlin]

dang

not sure though

im guessing its something like this

im probably wrong

[slow rover]

I'm almost sure it's wrong. The formula was pretty complex. I spent like 2 moths making it 10 years ago but it's lost 😄

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@slow rover Has your question been resolved?

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@slow rover Has your question been resolved?

[slow rover]

It's very easy to solve it using any programming language.
Very hard to create a math formula which might provide performance boost in calculations of very high numbers.

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[slim acorn]

Hey, so I've just been introduced to how the imaginary plane is at a 90deg angle from the real plane. i also know that imaginary numbers have a period of 4 when to exponents. This results in us having the following number line, from which we can make a circle. My question is, is it possible to deduce the equation for this circle?

[slim acorn]

and if so, what is it

[sharp flame]

I think you mean the imaginary axis is at 90 degrees to the real one

The two axes and the plane resulting is the complex plane

[slim acorn]

yes, the imaginary axis. is there a difference between the imaginary plane and the imaginary axis?

[sharp flame]

Anyways all that aside

[sharp flame]

Yeah

Imaginary "plane" doesn't really mean anything

we call the whole plane the complex plane

[slim acorn]

ah right, i get it

[sharp flame]

do you know what the modulus of a complex number is

[slim acorn]

yes i do

|z| = a^2 + b^2 if z = a + bi

[sharp flame]

Yeah precisely

And what does it mean geometrically?

[slim acorn]

its an application of the pythaogrean theorem, so we're finding the hypotenuse, which would also be the radius

OHHHHH

[sharp flame]

👀

[slim acorn]

so we have our radius, and we just need the coordinate for the center of our circle

[sharp flame]

Which is...

[slim acorn]

which in this is the origin

[sharp flame]

Yay

[slim acorn]

so its just x^2 + y^2 = |z|
how would I go about finding |z| here though? I dont know what my complex number is

or is this the maximum i can go

1+i could be the complex number, but so could -1 -i

[sharp flame]

well you know a point on the circle

[slim acorn]

wait, those would be the same if i take the modulus

[sharp flame]

neither of these lie on the circle

[slim acorn]

oh

I know the point on the circle (1,i)

[sharp flame]

that's not on the circle though

1, 0 is

[slim acorn]

is it not?

Oh right, I cant assume the x coordinate

because its on the imaginary axis

[sharp flame]

Yeah

[slim acorn]

not a regular cartesian graph

[sharp flame]

real part is 0

[slim acorn]

ahhhh right

[sharp flame]

it is actually

just the y coordinate is imgainary

and x coordinate is real

so i is just (0, 1)

[slim acorn]

yea, what i meant by that is that it has a bit more nuance than what im used to

[slim acorn]

not sure i get what you mean by this

[sharp flame]

If you overlay a regular xy plane on top of the complex plane

i and the point (0, 1) occupy the same space

[slim acorn]

I think I get it, would it be possible for you to draw a diagram please? I'm trying to doing so myself but not really sure where to start 😅

[sharp flame]

Your diagram is right

It's just that on the y axis you can't have the real part be 1

[slim acorn]

oh wait, i get it

[sharp flame]

I suggest you check out a video by Khan Academy on complex numbers, it would really help you

[slim acorn]

i misread (0,1) as (1,0), my bad

[sharp flame]

Ah

[slim acorn]

so z = a + bi
since im not moving horizontally at all, a = 0
I am moving 1 unit up, so b = 1 right?

[sharp flame]

yep

[slim acorn]

in which case z = 0 + 1i.
the modulus of which would just be 1

so the circle equation would just be x^2 + y^2 = 1? same as the regular unit circle?

[sharp flame]

yes

[slim acorn]

ah, thats a little underwhelming, but makes sense

[sharp flame]

on the complex plane the equation would simply be |z| = 1

[slim acorn]

was really hoping for something that looked a bit cooler, but oh well

[sharp flame]

reality is dissapointing

[slim acorn]

makes sense, thanks!

Yea, that should be it, appreciate the help, thanks again!

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[clever turret]

domain of: $\frac{\sqrt{x-1}}{x^2-4}$

[glossy valve]

Kondor

[clever turret]

i have a problem with this

$x^2-4\neq0$

[glossy valve]

Kondor

[clever turret]

$x^2\neq4$

[glossy valve]

Kondor

[clever turret]

$x\neq2$

[glossy valve]

Kondor

[clever turret]

then the root must be $\geq0$

[glossy valve]

Kondor

[polar valve]

thats not all, what about -2?

[clever turret]

$x\geq1$

[glossy valve]

Kondor

[clever turret]

$x\neq2 \ \ \ x\geq1$

[glossy valve]

Kondor

[clever turret]

-2 what?

[fallen hull]

(-2)^2 = 4

[clever turret]

where it comes out

-2^2=-4

[polar valve]

from x^2 not equal 4 you get x nt equal 2 oder -2

[fallen hull]

-2^2 = -4

(-2)^2 = 4

[clever turret]

I did it without brackets

[fallen hull]

yeah and when you substitute things you put brackets

[clever turret]

why?

[fallen hull]

idk

fact of the universe

[clever turret]

oh well I won't put them

easy

so did I do well?

[fallen hull]

thankfully -2 is >= 1 so you dont need to specify it not being equal to -2

it seems Ok to me

[clever turret]

?

Ok

is what i think

[sour torrent]

So what is the domain?

[clever turret]

this

[sour torrent]

so $x \in$

[glossy valve]

alee

[clever turret]

$x\in (x\geq1 \ \ \ x\neq2)$

[glossy valve]

Kondor

[clever turret]

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[lunar dagger]

For ex. 16, they want to know which values of a,b and c make the linear image sujrective

[lunar dagger]

Now i do not have a clue becaude normally i would day that de dim(Im) = dim(codomain), but since i don’t know what the codomain is i’m wondering how to solve this

[neat aspen]

can you translate exactly what it says

[lunar dagger]

Well

V and W are vectorspaces with α a basis for V and β a basis for W. The linear image is given by (see matrix). Wat values do a, b and c need to be so that L becomes surjective

[oak ruin]

The matrix has to be consistent for it to be surjective

i.e. you can't row reduce to a row of zeroes

[lunar dagger]

i do understand your example but what do u mean by 'being consisten' (we prob use other terminology in dutch)

[oak ruin]

Basically, when you row reduce it with a vector you're trying to map to, you don't end up with something like
0 0 0 | x
Because then there is no way to get x

[lunar dagger]

oh okay

so then that means that a != b, b !=c and a!=c

[lunar dagger]

but why is this?

[oak ruin]

Say the matrix is
$\begin{bmatrix}
1 & 0 & 0 \
0 & 1 & 0 \
0 & 0 & 0 \
\end{bmatrix}$.

Then there is no vector that maps to $\begin{bmatrix} 0 \ 0 \ 1 \ \end{bmatrix}$.

[glossy valve]

Ari

[oak ruin]

Also, the operator is surjective if and only if the determinant is not 0

[lunar dagger]

i don't think that's even a proposition, lemma or a theorem in my book

how would i be supposed to know this?

[oak ruin]

So $\det(A)=0 \iff A$ is not invertible.

[glossy valve]

Ari

[oak ruin]

And since $A$ is an operator, $A$ is invertible $\iff A$ is surjective.

[glossy valve]

Ari

[oak ruin]

And we know this by rank nullity theorem. A is invertible only if the null space is trivial, which would imply the range is the entire space

[lunar dagger]

ohhhh okay

thx

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[jolly silo]

So I found the derivative and found that the slope was 42 at x = 7.

Then solved for the scalar to be 400/sqrt(1813) that it gave me from the equation below.

I dont see how it's wrong at this point honestly.

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[nimble folio]

Can someone explain algebra to me? I need to know it for my examens and i'm a 7th grader. I don't understand anything from it.

[quartz gale]

Algebra is a large field of mathematics

So you might want to be more specific

[nimble folio]

Ow wait im gonna take my test where i got 40% 💀

[void nova]

Or ask in #prealg-and-algebra

[nimble folio]

2/7 X= 10/7

[leaden ermine]

,,\frac{2}{7}x = \frac{10}{7}?

[glossy valve]

𝔸dωn𝓲²s

[neat aspen]

monkey

[nimble folio]

why?

[neat aspen]

monkey d luffy

boku wa kozuki oden

[meager oracle]

word

[meager oracle]

why do u have the undergraduate math role if ure a 7th grader lol

[nimble folio]

Wdym

[hoary ember]

you can start by sending the 2/7 to the other side to isolate x

[meager oracle]

https://tenor.com/view/cat-funny-animals-package-gif-9243644

[nimble folio]

what does that mean?

[nimble folio]

so 2/7 X=

[meager oracle]

undergraduate is when ure in university

[nimble folio]

ow yh

idk

[hoary ember]

what

[meager oracle]

is this what u need help with?

[nimble folio]

yes?

[meager oracle]

what have you tried

[nimble folio]

X= 2/7 divided by 10/7 X= 2/7 times 7/10 X= 14/70 or 2/10

[meager oracle]

ok so

what would happen if u multiplied both sides by 7

[hoary ember]

pls dont use that format and your wrong

[hoary ember]

also this is almost correct
$$\frac{2}{7} \cdot x=\frac{10}{7}$$
send the $\frac{2}{7}$ to the other side
$$x=\frac{\frac{10}{7}}{\frac{2}{7}}$$

we can simplify to
$$x=\frac{10}{7} \cdot \frac{7}{2}$$

[glossy valve]

Skill_Issue

Skill_Issue

[hoary ember]

whoops sorry typo

[nimble folio]

its alright

i see my mistake

wait so is that the answer

?

[hoary ember]

yes

i mean you have to simplify it first

but yeah you will get the correct answer if you continue it

[nimble folio]

Thank you i see my mistake

how do i unclaim this space for other people?

[hoary ember]

.close

[nimble folio]

.close

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[fleet sage]

Could someone check my work here?

I'm just a little sussed out by the fact that the answer is zero

Feel like I did something wrong

Ping me if you respond pls thanks

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@fleet sage Has your question been resolved?

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@fleet sage Has your question been resolved?

[fleet sage]

bump

[lucid flower]

you’ll be better off asking in #multivariable-calculus maybe for visibility purposes/get it seen by people who like mv calc. if i see this tomorrow i might answer but i am too tired tonight

not to say that i don’t like mv calc

[fleet sage]

i'll try that thanks

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[quiet kraken]

I need help with this exact Question and two other ones. Here the first one: How old is the mother? Today she is 3 times as old as her daughter. 5 years ago she was 5 times her age.*

[ember ember]

Well

[hot herald]

start with introducing variables to represent their current ages

[ember ember]

Yes

[quiet kraken]

So D for Daughter and M for Mother

[hot herald]

try setting up equations for each statement, starting with

Today she is 3 times as old as her daughter

[quiet kraken]

So M = T•3

[hot herald]

T?

do you mean D?

[quiet kraken]

I mean D, sorry I was in German xD

[hot herald]

now try

5 years ago she was 5 times her age.*

[quiet kraken]

M-5=(D-5)•5 right?

[hot herald]

yes

you now have a system of equations, simplify/solve that with your preferred method of substitution or elimination

[quiet kraken]

For example?

[hot herald]

example of what

[quiet kraken]

Of how to solve this problem with my equations

[hot herald]

look up system of equations

[quiet kraken]

Should I watch this video

[hot herald]

yes

[quiet kraken]

My teacher wants it like this( just replace the t with d)

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@quiet kraken Has your question been resolved?

[quiet kraken]

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[silent lodge]

A boat can travel at a speed 20 km/h
h = (constant in magnitude) sailing in
calm water. The boat is traveling upstream. The current velocity of the
river is 5 km/h

The desired navigation route makes an angle of 60° with the perpendicular to the
bank of the river.
At what angle θ, measured relative to the bank, must the boat travel in order to follow the
desired cruising route.

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[crude wave]

I'm not sure how to start these kind of questions

[grave elm]

g(f(f(-2))) is what you're trying to find?

if so, then start by finding f(-2)

[crude wave]

so do i subsititute values from the graph for this?

[grave elm]

yep, basically

what's f(-2)?

read it from the graph

[crude wave]

is it 0?

[grave elm]

correct

so g(f(f(-2))) is just g(f(0))

because f(-2) is 0

so now you need to find f(0)

[crude wave]

will this be 8?

[grave elm]

f(0), not g(0)

[crude wave]

oo i see

so its like 6.2 or smth

[grave elm]

I'd suppose that it's exactly 6

[crude wave]

oo okk

[grave elm]

or close to 6

[crude wave]

now i subtitute in g(6)?

[grave elm]

we care just about the integer part anyway

[grave elm]

correct

[crude wave]

ooo oh ya

so is the answer 2?

[grave elm]

yes, i think so

[crude wave]

thank youu!!

[grave elm]

np

[crude wave]

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[crude wave]

.reopen

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[crude wave]

I've moved the terms to one side to get 6x^2-7xy-20y^2=0

I'm not quite sure how to factorise this

[grave elm]

no need to factorize anything

divide everything by x^2

[grave elm]

also, its -20y^2

[crude wave]

so root it??

[crude wave]

oh ye sorry!

i did use the quadratic formula to get x as 5/2 and -4/3

since its bigger than 0 can i substitute 5/2 in the eq to find y?

[grave elm]

x?

hmm

how can you get specific value for x 🤔

[crude wave]

i substituted the eq in the quadratic formula??

i think its wrong sorry

[grave elm]

note that youre finding just the ratio x/y

and you have 6x^2 = 7xy + 20y^2

try dividing everything by y^2

[crude wave]

so 6= sqrt 7xy+20 ??

[grave elm]

why sqrt?

7xy / y^2 = 7x/y

$\frac{6x^{2}}{y^{2}}=\frac{7xy+20y^{2}}{y^{2}}$

[glossy valve]

MethIsAlwaysRight

[crude wave]

OOO i seee thank youu

would i cross multiply now?

[grave elm]

nope

$6\left(\frac{x}{y}\right)^{2}-7\left(\frac{x}{y}\right)-20=0$

familiar?

[glossy valve]

MethIsAlwaysRight

[crude wave]

OOO its in the quadratic formm

[grave elm]

yep

now you can solve for (x/y)

and thats essentially what youre finding the whole time

[crude wave]

thank youu!!

[grave elm]

np

[crude wave]

wait i got 5/2 or -4/3 again

so the ratio of x:y would be 5:2?

[grave elm]

hmm

oh yeah

[grave elm]

well depends on what exaclty do they mean

x/y = 5/2

But yes

probably enter 5 : 2

[crude wave]

ohh okk

thank you again!!

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[grave elm]

np

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[hoary ember]

what is the limit as x approaches 0 of $\frac{sin(3x)}{x}$

[glossy valve]

Skill_Issue

[thick hedge]

try multiplying and dividing by 3

[hoary ember]

what

wdym?

[thick hedge]

$\lim_{x\rightarrow0}3\left(\frac{\sin\left(3x\right)}{3x}\right)$

[glossy valve]

ƒ(Why am. I here)=I don't Know

[hoary ember]

how do i find that?

[silk vault]

do you know subsitution in limits?

[hoary ember]

like subtitute 3x=u or smth?

oh is it 3

[silk vault]

yes

[silk vault]

yes.

lim(x-->0) of sinx/x is 1

[hoary ember]

thanks

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[hoary ember]

.reopen

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✅

[silk vault]

yes?

[hoary ember]

$\lim_{x\rightarrow0}\frac{x}{\tan(2x)}$

[glossy valve]

Skill_Issue

[silk vault]

um do you know d l hopital rules?

[thick hedge]

no need l'hopital

[hoary ember]

like the differentiate thing?

[silk vault]

yeah?

[thick hedge]

$\lim_{x\rightarrow0}\left(\frac{1}{2\frac{\tan\left(2x\right)}{2x}}\right)$

[glossy valve]

ƒ(Why am. I here)=I don't Know

[silk vault]

yes..?

is this is a known thing?

tanx/x?

[thick hedge]

I think so

[silk vault]

i havent seen it

but maybe

[thick hedge]

depends in the region, I guess

[hot herald]

you can convert it down to get sin(x)/x again

[silk vault]

oh yeah true

[silk vault]

subsitute tan(2x) with sin2x/cos2x

[hoary ember]

$\lim_{x\rightarrow\inf ^{\plus}}\sqrt{x^2-5x}-\sqrt{x^2-3}$

[glossy valve]

Skill_Issue
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[hoary ember]

oh yeah i got it

[silk vault]

you can multiply and divide this with (sqrt(x^2-5x)+sqrt(x^2-3))

[hot herald]

\infty for infinity

[hoary ember]

$\lim_{x\rightarrow\infty^{\plus}}\frac{-5x+3}{\sqrt{x^2-5x}+\sqrt{x^2-3}}$

[glossy valve]

Skill_Issue
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[silk vault]

yes

[hoary ember]

i got this

[silk vault]

now you take a common factor of x from the numerator and a common factor of x^2 in the roots of the denominator

[hoary ember]

i dont understand

[silk vault]

well -5x+3

is x(-5+3/x)

right?

@hoary ember

[hoary ember]

yea

yeag

[silk vault]

and x^2-5x= x^2 (1-5/x) right?

[hoary ember]

yesh

[silk vault]

and x^2-3 is x^2 (1-3/x^2) right?

[hoary ember]

yes

-5/2?

aww nope

[silk vault]

yes.

the result

[hoary ember]

its still incorrect apparently?

[silk vault]

it is?

[silk vault]

well that has +5x not -5x

so it will end up as 5/2

[hoary ember]

bruh

alr ty

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[silk vault]

np

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[waxen quail]

can someone tell me how they got this answer for b please?

[waxen quail]

nvm nvm

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[rustic dagger]

i dont understand the graphing part

if x=2

it should be 15

[fallen hull]

why

[rustic dagger]

why what

[fallen hull]

why is it 15