A box with an open top has vertical sides, a square bottom, and a volume of 256 cubic meters. If the box has the least possible surface area, find its dimensions.

Answer 1

Volume = 256 m³

Let base = x by x m
height = 256/x²
Surface area = x²+4(x)(256/x²)
=x²+1024/x

differentiate with respect to x and equate to zero for extremum:
2x-1024/x²=0
x=8

So the box size is 8x8x4 high

Note: I have assumed that the base area is a square because the square is the shape with least perimeter.

We could assume different dimensions and solve the problem by Lagrange multipliers, but the result will be the same.

Answer 2

thnx so much :)

Answer 3

How did you get the x^2+4x part in the surface area calculation

Answer 4

To the first Anon,
The formula for Surface Area is 2lw+2lh+2hw
Since the base is square, length and width are equal. Since the box has no top, you change 2lw (which is the surface area of the base and top) into lw. You can turn both of these terms into one term instead, in this instance x. With this in mind, you get: (x)(x)+2(x)(h)+2(h)(x)
You can simplify this into x^2+4xh
We solved h to equal 256/x^2 in a previous step, so we can plug that in here too, leaving us with
x^2+4(x)(256/x^2)