Make a box with a square base using only 250 square feet of material.

Question

what dimensions for the box will produce a maximum valume?
what is the maximym volume?

Answer 1

I will assume an open box with no lid

base --- x by x
height --- y
x^2 + 4xy = 250
y = (250-x^2)/(4x)

V = x^2y = x^2(250-x^2)/(4x)
= 250x/4 - x^3/4
dV/dx = 250/4 - (3/4)x^2 = 0 for a max V
250 = 3x^2
x^2 = 250/3
x = √250/√3 , y = (250-250/3)/(4√250/√3)
x = appr. 9.129 , y = appr. 4.564

max volume = x^2y = 380.362

check:
when x = 9.1 , V = 380.357
when x = 9.2 , V = 380.328 both are less than my answer of 9.1229

Answer 2

answer is supposed to be x = 6.45.
h = [125 - 6.45^2] / 2(6.45) = 6.4649

I don't see where this is from.

Answer 3

Remember I said that I will assume there is no top, as is the case in the majority of this type of question.

Let's try it with a top,
then my opening equation would be
2x^2 + 4xy = 250
x^2 + 2xy = 125
y = (125 - x^2)/(2x)
then follow the same steps to get their answer

If there is a top, then the best volume will be obtained when you build a cube, that is, the height is also x
then of course all 6 faces would be squares, and you would have
6x^2 = 250
x^2 = √(250/6) = 6.45497