A box is to be constructed from a sheet of cardboard that is 20 cm by 60 cm, by cutting out squares of length x by x from each corner and bending up the sides. What is the maximum volume this box could have?

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A box is to be constructed from a sheet of cardboard that is 20 cm by 60 cm, by cutting out squares of length x by x from each corner and bending up the sides. What is the maximum volume this box could have?

Mathematics
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|dw:1436828452402:dw|
if we make the cuts and fold up the box, we get something like this |dw:1436828568511:dw|
Okay. What next? I know Volume would be lwh --- so x(20-2x)(60-2x)?
correct
So, then what?
ok so if you graph x(20-2x)(60-2x) with something like desmos, you get this (see attached)
Okay?
So 2525 is the max volume?
desmos is saying that the max point is (4.514, 2525) so x = 4.514 and the max volume is 2525 yes
Thank you!
no problem
So what if you wanted to work it out by hand? Just wondering..
to do so by hand, you would need to know calculus (specifically, derivatives)
I do haha. I did it originally using derivatives and my teacher yelled at me and marked it wrong
such a strange thing to be yelled at for lol
I'm guessing you're in precalculus and the teacher wanted you to use technology like a graphing calculator
I'm actually in calculus and my sister taught me derivatives
ok let's use derivatives then
first expand out x(20-2x)(60-2x) to get x(20-2x)(60-2x) = x(1200 - 160x + 4x^2) = 4x^3 - 160x^2 + 1200x
then you apply a derivative to 4x^3 - 160x^2 + 1200x to get y = 4x^3 - 160x^2 + 1200x dy/dx = 3*4x^(3-1) - 2*160x^(2-1) + 1*1200x^(1-1) dy/dx = 12x^2 - 320x + 1200
Ya, I got that far. Then quadratic?
Now you set the derivative 12x^2 - 320x + 1200 equal to zero and solve for x 12x^2 - 320x + 1200 = 0 x = 22.15250437 or x = 4.514162294 I skipped a bunch of steps since I'm sure you know the quadratic formula very well at this point
Then plug them back in for the y
since the width is 20-2x, this means that 20-2x must be positive 20 - 2x > 0 20 > 2x 2x < 20 x < 10 so x is both positive, x > 0, and less than 10. Put together, 0 < x < 10 is the domain restriction here. This is visually confirmed in that desmos graph above
because 0 < x < 10, we can ignore x = 22.15250437
I find it extremely frightening that you just completed the problem so quickly going the long way
I've had lots of practice
Maybe one day I will be able to do that
you will with enough practice
Again, thank you for your help
no problem

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