Optimization | Mathematics | Page 1

steep spruce Dec 13, 2023, 5:22 AM

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So, i'm doing some differentation practice right now

terse copperBOT Dec 13, 2023, 5:22 AM

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steep spruce So, i'm doing some differentation practice right now

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steep spruce Dec 13, 2023, 5:22 AM

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Oh and I cant send images

plain pelican Dec 13, 2023, 5:22 AM

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you can man

steep spruce Dec 13, 2023, 5:22 AM

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thats lovely

plain pelican Dec 13, 2023, 5:22 AM

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trust me you can do it

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everyone has image perms in this channel

pliant solstice Dec 13, 2023, 5:22 AM

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comeone bro

steep spruce Dec 13, 2023, 5:22 AM

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can you not clipboard it?

plain pelican Dec 13, 2023, 5:23 AM

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✊

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come on bro

pliant solstice Dec 13, 2023, 5:23 AM

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i believe in you homie

plain pelican Dec 13, 2023, 5:23 AM

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everyone can send images in this channel

steep spruce Dec 13, 2023, 5:23 AM

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lemme do it like this

pliant solstice Dec 13, 2023, 5:23 AM

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look

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@steep spruce you there?
'

steep spruce Dec 13, 2023, 5:24 AM

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hold on

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just downloading the image

pliant solstice Dec 13, 2023, 5:24 AM

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alr

steep spruce Dec 13, 2023, 5:25 AM

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doesn't let me ctrl v it for some reason

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either way

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so with this optimization question

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i'm not entirely sure how to approach in solving it

pliant solstice Dec 13, 2023, 5:26 AM

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10a?

steep spruce Dec 13, 2023, 5:26 AM

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yeah

pliant solstice Dec 13, 2023, 5:26 AM

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okay well

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tell me what the height of the box will be

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it will be x

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the width will be 28 - 2x

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and the length will be 55 - 2x

steep spruce Dec 13, 2023, 5:28 AM

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just x

pliant solstice Dec 13, 2023, 5:28 AM

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multiply those three out and i'm pretty sure you get that cubic

steep spruce Dec 13, 2023, 5:28 AM

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oh you sent it

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yeah I get that 1540x is the result of h x w x l

plain pelican Dec 13, 2023, 5:29 AM

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no

pliant solstice Dec 13, 2023, 5:29 AM

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huh

plain pelican Dec 13, 2023, 5:29 AM

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you know how the volume is l x b x h

pliant solstice Dec 13, 2023, 5:29 AM

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yea...

steep spruce Dec 13, 2023, 5:29 AM

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oh yeah you're right ignore me

plain pelican Dec 13, 2023, 5:29 AM

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now you know l is 55-2x, b is 28-2x and h is x

plain pelican Dec 13, 2023, 5:29 AM

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steep spruce oh yeah you're right ignore me

yeah

pliant solstice Dec 13, 2023, 5:29 AM

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plain pelican now you know l is 55-2x, b is 28-2x and h is x

literally what i said

late bisonBOT Dec 13, 2023, 5:30 AM

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wolfqz

plain pelican Dec 13, 2023, 5:30 AM

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pliant solstice literally what i said

i just noticed they did NOT understand

pliant solstice Dec 13, 2023, 5:31 AM

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bro

steep spruce Dec 13, 2023, 5:31 AM

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yeah so multiplying x(28-2x)(55-2x) = 4x^3 - 166x^2 + 1540x

plain pelican Dec 13, 2023, 5:31 AM

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steep spruce yeah so multiplying x(28-2x)(55-2x) = 4x^3 - 166x^2 + 1540x

yeah

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the other parts are straightforward

steep spruce Dec 13, 2023, 5:32 AM

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well thats the thing I know how to find the derivative of 4x^3-166x^2+1540x

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but I don't know how to find the maximum of things

pliant solstice Dec 13, 2023, 5:32 AM

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okay i gotchu

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find the derivative now

steep spruce Dec 13, 2023, 5:34 AM

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4x 3 = 12
-166 x 2 = 322
1540x = 1540

12x^2-322x+1540

plain pelican Dec 13, 2023, 5:34 AM

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i dont think he'll get the intuition over text

steep spruce Dec 13, 2023, 5:34 AM

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oh whoops

pliant solstice Dec 13, 2023, 5:34 AM

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bro shutup

steep spruce Dec 13, 2023, 5:34 AM

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322

pliant solstice Dec 13, 2023, 5:34 AM

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@plain pelican

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go

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lemme do this

plain pelican Dec 13, 2023, 5:35 AM

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if he doesnt know how to find a maximum it would be really hard to actually understand the why behind the process of finding maxima

pliant solstice Dec 13, 2023, 5:35 AM

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we don't need two ppl helping

pliant solstice Dec 13, 2023, 5:35 AM

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plain pelican if he doesnt know how to find a maximum it would be really hard to actually unde...

i got it

plain pelican Dec 13, 2023, 5:35 AM

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alright i am watching

steep spruce Dec 13, 2023, 5:35 AM

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alright

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so yeah derivative is 12x^2-322x+1540

pliant solstice Dec 13, 2023, 5:36 AM

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okay good

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now let's imagine the graph of the function of the box cubic

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so y = 4x^3 - 166x^2 + 1540x

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another thing

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we know that 0 < x < 28

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as if x was bigger than 28 it would go past the whole box

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okay so

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you have the graph visualized?

steep spruce Dec 13, 2023, 5:38 AM

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when I plot the graph its just a straight line on the y axis no?

pliant solstice Dec 13, 2023, 5:38 AM

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y = 4x^3 - 166x^2 + 1540x

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plot that

steep spruce Dec 13, 2023, 5:38 AM

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i get a straight line

pliant solstice Dec 13, 2023, 5:38 AM

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what

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its a cubic

steep spruce Dec 13, 2023, 5:38 AM

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🤷

pliant solstice Dec 13, 2023, 5:38 AM

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send screeshot

steep spruce Dec 13, 2023, 5:38 AM

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I put word for word what you sent

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its an in person calculator

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so 1 sec

pliant solstice Dec 13, 2023, 5:40 AM

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oh i seee

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yea ofc

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because the value of 1540x is so much larger than the rest of the values when x is small

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it should look like a straight line

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but in fac

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fact*

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the slope is actually just a very very high number

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not quite infinity

steep spruce Dec 13, 2023, 5:41 AM

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I guess my y axis isn't big enough

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lemme increase the window size

pliant solstice Dec 13, 2023, 5:41 AM

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no it's fine

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just use simple intuition

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if it's a cubic

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it's obviously not a straight line

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but for example at x = 0

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the slope is 1540

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and so in a calculator without a very big zoom-in it would look like a vertical line

steep spruce Dec 13, 2023, 5:42 AM

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jesus christ it doesn't curve until well past y = 2000

pliant solstice Dec 13, 2023, 5:43 AM

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yea its a big function

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okay so

plain pelican Dec 13, 2023, 5:43 AM

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it does lol

steep spruce Dec 13, 2023, 5:43 AM

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slope should peak around 4

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k

pliant solstice Dec 13, 2023, 5:43 AM

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bro shutup wolf

steep spruce Dec 13, 2023, 5:43 AM

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is what im getting

pliant solstice Dec 13, 2023, 5:43 AM

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twist

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imagine the function

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so basically

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in order for it to be a maximum

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actually nvm

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think of it this way

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:

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if at a certain point (a, b) it is a local maxima, what are you given about the points right next to that

steep spruce Dec 13, 2023, 5:45 AM

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the points right next to point (a,b)?

pliant solstice Dec 13, 2023, 5:45 AM

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yea

steep spruce Dec 13, 2023, 5:46 AM

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a - 1 and a + 1?

pliant solstice Dec 13, 2023, 5:46 AM

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no the points like right next to a, b

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have you done infinitecimals yet

steep spruce Dec 13, 2023, 5:46 AM

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what

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no idea what that is

pliant solstice Dec 13, 2023, 5:47 AM

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okay well let's think

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the points right next to a, b

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let's call one of them a + something really small, c

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you know that because b is a local minimum

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c has to be smaller than b, right

steep spruce Dec 13, 2023, 5:49 AM

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yeah

pliant solstice Dec 13, 2023, 5:49 AM

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and that's true about a - something really small, c

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c still has to be smaller than b, right

steep spruce Dec 13, 2023, 5:50 AM

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yes

pliant solstice Dec 13, 2023, 5:50 AM

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so then you know that the graph as x approaches a slowly grows, then as soon as it hits a it starts falling

steep spruce Dec 13, 2023, 5:51 AM

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yeah

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so in this situation a = x yeah?

pliant solstice Dec 13, 2023, 5:51 AM

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yes

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so what does that tell you about the derivative of f(x) at x = a?

steep spruce Dec 13, 2023, 5:52 AM

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well if you set the equation = 0 and solve for x you would find the x value yeah?

pliant solstice Dec 13, 2023, 5:52 AM

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exactly

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12x^2-322x+1540

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so when that equals 0, it's a local minimum

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can you find its roots?

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use desmos or wolframalpha or smth

steep spruce Dec 13, 2023, 5:54 AM

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x1 = 6.23
x2 = 20.61?

pliant solstice Dec 13, 2023, 5:55 AM

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okay so both of those can either be local minimums or local maximums, right?

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as the same logic to the maximum can apply to the minimum

steep spruce Dec 13, 2023, 5:55 AM

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yeah

pliant solstice Dec 13, 2023, 5:56 AM

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so now graph the graph

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and see which one of them makes sense

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and you're done!

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the answer has to be one of those

steep spruce Dec 13, 2023, 5:57 AM

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you mean graph 4x^3 - 166x^2 + 1540x

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find where 6.23 and 20.61 is

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and see which one shows me to be the peak of the function

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i think i've done my math wrong here

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cause the highest the y value ever gets is 4129

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and thats not on x = 6.23

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its on x = 5.894

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can someone confirm this with me?

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@pliant solstice ?

#Optimization