Calculus optimization: given surface area, want largest volume of an open-top box

[@WinglessPenguin]

You are quite literally my hero

[@GirishManjunathMusic]

x² + 4xy = 1200 4xy = 1200 - x² y = (1200 - x²)/4x Maximise x²y: d/dx x²y = d/dx x²(1200 - x²)/4x = d/dx 300x - ¼x³ = 300 - ¾x² to maximise, set to 0 to find critical value: = 3(10 - ½x)(10 + ½x) = 0 x = ±20 x = -20 doesn't make sense for a physical object, so let us take the positive value: x = 20 units. y = (1200 - x²)/4x = (1200 - 400)/80 = 800/80 y = 10 units. volume = x²y = 20·20·10 = 4000 cubic units.

[@phoenig9302]

Tomorrow is mature exam in basic maths and 11.05 is extended maths in Poland. This is typical optimisation question in extended exam for 7 of 50 points

[@prajwalgowda1375]

there are more analytical problem in application of derivatives than the one which ur solving 😊.....ur making it look so easy ...

[@田村博志-z8y]

This is 4th time of AM-GM. Let x^2 + 4xy = a^2 be fixed. We note x^2 + 2xy + 2xy = a^2. By AM-GM inequality we have ( x^2・2xy・2xy )^( 1/3 )

[@bcholio]

How would you do it if the y value wasn’t constant? Like one side was taller than the other? The problem didn’t specify a constant height. Could you maximize volume more?

[@RamAurelius]

In that case I think you'd need to develop general equations for the volume and surface area, since we wouldn't be able to take the shape to be a rectangular prism. Y itself would have to be some function in terms of X, so I feel like it would start to become more like a 3D analytical geometry problem. Would love to see it attempted though, it sounds like it could be an interesting process!

[@Γιώργος-η8γ]

Can u make another video where u solve this kind of problem using anadelta? It would be great! P.S:Nice video once more!!!

[@danielkovacs6809]

Hi! I recently watched one of your older videos about the summation of 1/n^3 as n goes from 1 to infinity. By trying on the calculator, i think, I found a nice form of that approximation for the sum. It's 1,7/sqrt(2). What do you think? :)

[@田村博志-z8y]

Setting a := Σ_{ n = 1 }^{ ∞ } 1/n^3 the calculator shows that a = 1.20205690… 1.7/√2 = 1.20208152… How much error is between a and 1.7/√2 by your method ?

[@danielkovacs6809]

@@田村博志-z8y Around 0,00002463.

[@田村博志-z8y]

@@danielkovacs6809 I don't mean that. How did you get 1.7/√2 ? No method but only using the calculator ?

[@littleretroship6403]

Ok so if the box has an open top and a square base let’s set the variables. Let’s say that a is the side of the square. So we’ll have 4 other rectangles let’s say their heights are b. So a^2 + 4ab = 1200. And the volume will be a^2 * b. Now I believe we put it into 1 variable and plug it into the volume function. So ig I’ll solve for b and later I’ll derive with respect to a. So factor: a(a+4b)=1200 and divide by a: a+4b=1200/a. Now I subtract and divide I think. 4b=(1200/a)+a and now: b=((1200/a)+a)/4. Ok so the function (once simplified) will be (let’s say V(a) is the volume) V(a)=(a^3)/4 + 300a. Ok and the derivative is just V’(a)=(3a^2)/4 + 300. Now we set it to 0. 0=(3a^2)/4 + 300. We can now solve for a. -300=(3a^2)/4 and now -400=a^2. Uh oh. Well I messed up. I should’ve chose b maybe or i did smt wrong along the way. Lemme redo this Ah my mistake was I added a instead of subtracting Edit: ok here we go 4ab=1200-a^2 b=(1200-a^2)/4a V(a)=a^2 * ((1200-a^2)/4a) And ig I’ll foil V(a)= (1200a^2 - a^4)/4a And simplify again… V(a) = 300a - (a^3)/4 And derive V’(a)=300 - (3a^2)/4 And set to 0 0=300 - 3/4 * a^2 And subtract -300= -3/4 * a^2 And divide by 3/4 -400 = -a^2 And now I can multiply by -1 and sqrt 20 = a And finally I can find b and the volume b=(1200-a^2)/4a So b= (1200 - 400)/80 And b= 800/80 Here we go: b = 10 10*20*20=4000 So the largest possible volume for the box is 4000cm^3 Edit again: I’ve made a mistake but I learned from it and I found it Edit: forgot about the negative square root but I don’t think it matters in this bc I’m pretty sure in the real world that boxes don’t have negative volumes and dimensions lol

[@GR47]

How is this comment is 1 day ago

[@dfsgjlgsdklgjnmsidrg]

sorry im about to unsubscribe if there is uploaded another cube the box question video. do something like polynominal regression with an alpha smoothing