Question

I don't really understand optimization problems and i need help.

A box with a square base and open must have a volume of 32,000 cm^3. Find the dimensions of the box that minimize the amount of material used.

Answer 1

I want to minimize the amount of material used. So we need to find minimum point, in which the derivative of the area with respect to one of the dimensions is 0, since that's the only time for minimum points to occur:
Let the width of box = x, height of box = y
Volume, V = yx^2
y = 32000/(x^2) ----(1)
Equation (1) represents the constraints defined by the question.
Area, A = x^2 <for square base> + 4xy <for the four sides> ----- (2)
We need to try to force (2) to be the minimum, but first, plug in (1) into (2):
A = x^2 + 128000/x
Differentiating wrt x,
A' = 2x - 128000/(x^2)
For minimum A, A' = 0
2x^3 - 128000 = 0
2x^3 = 128000
x^3 = 64000
x = 40
A'' < 0 (I'm not going to show the working for this but it's just differentiating A' wrt x once more)
y = 32000 / (40^2) = 20
The dimensions are 40cm x 40cm x 20cm

Answer 2

"Thank You

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Luigi0210"