If the box is square with side length x, the surface area is
a = xy+2yz+2xz
= x^2 + 2z(2x)
= x^2 + 4x(32/(x^2))
= x^2 + 128/x
This will be minimized when the derivative with respect to x is zero.
2x - 128/x^2 = 0
2x^3 - 128 = 0
x^3 = 64
x = 4
A square box that is 4 ft by 4 ft and 2 ft deep will have minimum surface area.
Its area will be 16 ft^2 + 32 ft^2 = 48 ft^2.
If the box is 4.1 ft by 3.9 ft by 2.00125 ft, the area will be ≈ 48.01 ft^2. Thus, it looks like the box should be square for minimum surface area.
a = xy+2yz+2xz
= x^2 + 2z(2x)
= x^2 + 4x(32/(x^2))
= x^2 + 128/x
This will be minimized when the derivative with respect to x is zero.
2x - 128/x^2 = 0
2x^3 - 128 = 0
x^3 = 64
x = 4
A square box that is 4 ft by 4 ft and 2 ft deep will have minimum surface area.
Its area will be 16 ft^2 + 32 ft^2 = 48 ft^2.
If the box is 4.1 ft by 3.9 ft by 2.00125 ft, the area will be ≈ 48.01 ft^2. Thus, it looks like the box should be square for minimum surface area.
