Multivariable calc help needed asap plz | Mathematics | Page 1

orchid streamBOT Nov 10, 2023, 7:48 AM

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raw igloo Nov 10, 2023, 7:48 AM

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the gradient P(s,t) = (6000e^t - 4000s, 6000se^t)

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This is the system of equations I have set up that I need to solve:

6000e^t - 4000s = 2 * lambda * s

6000se^t = lambda*e^t

s^2 + e^t = 11/3
I have no idea how to proceed with solving this after thinking about this for a long time...

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Please can someone provide the answer with a detailed explanation...

raw igloo Nov 10, 2023, 8:09 AM

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@grave tide u mind taking a look at this when ur awake?

grave tide Nov 10, 2023, 9:58 AM

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I assume P(s,t)=6000se^t - 2000s^2 + C
L(s,t,λ)=P(s,t)+λ(s^2+e^t-11/3)
∂L/∂s = 6000e^t - 4000s + 2λ = 0
∂L/∂t = 6000se^t + λe^t = 0
∂L/∂λ = s^2 + e^t - 11/3 = 0
then you resolve your values, I got t=ln(8/3), s=1, λ=-6000 as the only valid sol

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@raw igloo

raw igloo Nov 10, 2023, 6:24 PM

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grave tide I assume P(s,t)=6000se^t - 2000s^2 + C L(s,t,λ)=P(s,t)+λ(s^2+e^t-11/3) ∂L/∂s = 6...

Where did u get assume P(s,t) = 6000se^t - 2000s^2 + C from?

grave tide Nov 10, 2023, 6:42 PM

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the original function

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you said your gradient was (6000e^t -4000s, 6000se^t) so I derived the original function from that

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@raw igloo

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idk whether you had a constant in the original function, but that doesn't affect the Lagrange multipliers

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cool

raw igloo Nov 10, 2023, 6:45 PM

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That was the original function

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But why r we using the original for lagrange here because they only care about it perpendicular to the boundary

prime apexBOT Nov 10, 2023, 6:46 PM

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apex predator

raw igloo Nov 10, 2023, 6:46 PM

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Which is just this:

grave tide Nov 10, 2023, 6:47 PM

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I didn't read the question 💀

raw igloo Nov 10, 2023, 6:48 PM

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They only care about gradient P = lambda*gradientg

grave tide Nov 10, 2023, 6:48 PM

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yeah shit

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I wasted my time 💀

raw igloo Nov 10, 2023, 6:48 PM

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raw igloo This is the system of equations I have set up that I need to solve: 6000e^t - 4...

I got this system, idk if it's right...

grave tide Nov 10, 2023, 6:48 PM

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I should consider reading the question at some point

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yeah your system looks fine

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you can divide out e^t in the second part as e^t≠0

raw igloo Nov 10, 2023, 6:49 PM

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I did this, but this seems very suspect because my numbers I got in the end r so weird lmao

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U mind checking if what I did is right?

grave tide Nov 10, 2023, 6:50 PM

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6000s=lambda
6000e^t - 4000s = 12000s² and e^t=11/3 - s²
22000-6000s²-4000s=12000s²

raw igloo Nov 10, 2023, 6:51 PM

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Where did 22000 come from?

grave tide Nov 10, 2023, 6:51 PM

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6000(11/3)

raw igloo Nov 10, 2023, 6:51 PM

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Nvm

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Ye

grave tide Nov 10, 2023, 6:52 PM

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then you can use the 6000s=lambda again

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getting an egregious cubic

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0=72000s³+6s²+4s-22

raw igloo Nov 10, 2023, 6:53 PM

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How do u have 12000s^2* lambda tho? Because we have 2 * lambda * s on the right side
So it should just be 2*(6000s)*s which is just 12000s^2 no?

grave tide Nov 10, 2023, 6:54 PM

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where is lambda going?

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oh yeah I replaced it

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mb

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then it's only a square

raw igloo Nov 10, 2023, 6:54 PM

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So there should be no cubic cuz we just have, 22000 - 6000s^2 - 4000s = 12000s^2

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Yeah

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And then we just apply quadratic formula

grave tide Nov 10, 2023, 6:55 PM

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9s²+2s-11=0

raw igloo Nov 10, 2023, 6:56 PM

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Uh

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How did u get that?

grave tide Nov 10, 2023, 6:56 PM

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then you can use t=ln(11/3-s²)

grave tide Nov 10, 2023, 6:57 PM

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raw igloo How did u get that?

18000s²+4000s-22000=0
18s²+4s-22=0
9s²+2s-11=0

raw igloo Nov 10, 2023, 6:57 PM

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Wow

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And that gives x = 1 and x = -11/9

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But we ingore the negative

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Since the question said only care about s,t >= 0

grave tide Nov 10, 2023, 6:58 PM

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yes

raw igloo Nov 10, 2023, 6:58 PM

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Bro

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You are actually such a legend

grave tide Nov 10, 2023, 6:58 PM

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ln(8/3) and SUDDENLY

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you have EXACTLY

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what I CALCULATED

raw igloo Nov 10, 2023, 6:58 PM

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😂

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Yes just different method

grave tide Nov 10, 2023, 6:59 PM

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rumour has it: if your Lagrange multipliers exist on your boundaries then IT'S THE SAME AS BEING PERPENDICULAR!!!!!!

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anyways, I was waiting to share that

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learning experience

raw igloo Nov 10, 2023, 6:59 PM

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Bro, u are actually a legend. Like actually next lvl talent

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I was on the other math discord and not 1 out of the 70,000 people

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Knew multivariable calc

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Legit no one

grave tide Nov 10, 2023, 6:59 PM

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lmao

raw igloo Nov 10, 2023, 7:00 PM

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They were all Indians swearing in their language and saying that I'm asking bs questions 😂

grave tide Nov 10, 2023, 7:00 PM

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lmao

raw igloo Nov 10, 2023, 7:01 PM

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Ok so now that I have that one point of s = 1 and t = Ln (8/3). They asked me to calculate the profit at that point, so I just plugged it into the original profit equation.

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.

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Which is this

grave tide Nov 10, 2023, 7:01 PM

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yeah

raw igloo Nov 10, 2023, 7:01 PM

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They then asked me this final thing:

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Max value is just the value of P at s = 1 and t = ln(8/3) right?

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Because Lagrange gives us the max/min value?

grave tide Nov 10, 2023, 7:02 PM

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that's not the global max

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that's the max given the contract s²+e^t=11/3

raw igloo Nov 10, 2023, 7:03 PM

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Yeah I think that's what they want

grave tide Nov 10, 2023, 7:03 PM

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if it requires you continue with the constraint then yeah

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positive profit isn't possible

raw igloo Nov 10, 2023, 7:04 PM

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Yeah ok

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That's cuz the max value

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Is negative right?

grave tide Nov 10, 2023, 7:04 PM

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max is negative, yes

raw igloo Nov 10, 2023, 7:04 PM

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Wow.

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Change ur name to Daddy Apex Predator asap

grave tide Nov 10, 2023, 7:05 PM

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wait no

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max is positive

raw igloo Nov 10, 2023, 7:05 PM

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Wut

grave tide Nov 10, 2023, 7:05 PM

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one sec I might have calculated wrong

raw igloo Nov 10, 2023, 7:06 PM

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P(1, ln(8/3)) = 6000(1)e^(ln(8/3)) - 2000 - 120000

grave tide Nov 10, 2023, 7:06 PM

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ln(8/3)=t s=1
6000(8/3)-14000=2000

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it's not 120k it's 12k

raw igloo Nov 10, 2023, 7:07 PM

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Ah

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It is positive indeed

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The max profit is around 1986

grave tide Nov 10, 2023, 7:08 PM

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How'd you get that?

raw igloo Nov 10, 2023, 7:10 PM

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6000(1)e^(ln(8/3) = 16000

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16000 - 2000(1)^2 - 12000

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= 2000

grave tide Nov 10, 2023, 7:10 PM

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so Why'd you get 1986?

raw igloo Nov 10, 2023, 7:10 PM

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Nvm I musta misclicked something earlier

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Lmao

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Mb

grave tide Nov 10, 2023, 7:10 PM

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ok

raw igloo Nov 10, 2023, 7:11 PM

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Ok nice nice

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I can't thank u enough for ur help, rlly do appreciate it. Ur the only person who explains everything so clearly and concisely and correctly everytime

grave tide Nov 10, 2023, 7:12 PM

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np

raw igloo Nov 10, 2023, 7:46 PM

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+close

#Multivariable calc help needed asap plz