Question 1097730
<br>Part A:<br>
Let x be the length of a side of the square base, and let h be the height.  The volume is then
{{{V = x^2h = 867}}}
which means
{{{h = 867/x^2}}}<br>
The surface area is comprised of the square top and bottom and four sides, each x by h:
{{{S(x) = 2x^2 + 4xh = 2x^2 + 4x(867/x^2) = 2x^2+3468x^(-1)}}}<br>
Part B:<br>
To minimize the surface area, we need the derivative S'(x) to be zero:
{{{4x- 3468x^(-2) = 0}}}
{{{4x^3-3468 = 0}}}
{{{x^3 = 867}}}<br>
The box with a volume of 867 cubic inches with the smallest surface area is a cube with edge length equal to the cube root of 867.<br>
Note that the calculus answer is what we would expect, since a box with fixed volume and minimum surface area is a cube.