Expert: Ahmed Salami - 3/7/2011

Question
your have a box with a square base, suppose there is no top (only 5 faces) if the volume must be 32 cubic ft what dimensions will minimize the surface area of the box?

Answer
Hi Amy,
Let the length of the square base be x and the height be h, then the surface area of the box is
A = 4hx + x²
The volume is
V = hx² = 32
h = 32/x²
The surface area can now be rewritten as
A = 4(32/x²)x + x²
= 128/x + x²
and this is optimal when dA/dx = 0
Now,
dA/dx = -128/x² + 2x
equating to zero
2x - 128/x² = 0
2x = 128/x²
x³ = 64
x = 4 ft

h = 32/4²
= 32/16
= 2 ft

Regards