Question

A box with a square base and open top must have a volume of 202612 cm3. We wish to find the dimensions of the box that minimize the amount of material used.

(Round your answer to the nearest tenthousandths if necessary.)


Length =
Width =
Height =

Answer 1

The Volume of a box with a square base x by x cm and height h cm is V=x2h

The amount of material used is directly proportional to the surface area, so we will minimize the amount of material by minimizing the surface area.

The surface area of the box described is A=x^2+4xh

We need A as a function of x alone, so we'll use the fact that
V=x2h=202612 cm^3

which gives us h=32,000x2, so the area becomes:

A=x^2+4x(202612/x^2)=x^2+810448/x

We want to minimize A, so

A'=2x−810448/x2=0 when (2x^3−810448)x^2=0

Which occurs when x^3-405224 or x=74

The only critical number is x=74 cm.

The second derivative test verifies that A has a minimum at this critical number:
A''=2+1620896/x3 which is positive at x=74

The box should have base 74 cm by 74 cm and height 37 cm.