Calculus optimization problem (minimizing material)

Question

A box with a square base and an open top must have a volume of 256m^3. Find the dimensions of the box that minimize the amount of material used. 

Answer 1

Hi Alex,

 

The volume of a box with a square base is given by V=x²h where x is the length of a side of the square base. The fact that the volume must be 256 allows us to establish a relationship between x and h, i.e. 256 = x²h, so h = 256/x². Use that later.

 

We need a function for the surface area.

 

S = x² + 4xh   (area of base plus area of 4 sides--no top).

 

We have h in terms of x, so S = x² + 4x(256/x²). Simplify, find first derivative, setting equal to zero. Find critical numbers (x=?) That value of x will determine h too. Convince yourself that the value of x generates a minimum surface area (2nd derivative test or similar).

 

You should be able to take if from there, but let me know if not.