surface area

Question

A company will make a cereal box with whole number dimensions and a volume of 100 cubic centimeters. If cardboard costs $0.05 per 100 square centimeters, what is the least cost to make 100 boxes?

Answer 1

If the dimensions of the box are x,y,z the volume is V=xyz.

 

The surface area will be A=2(xy + yz +xz)

 

A can then be written as:

 

A = 2(xy + y[100/xy] + x[100/xy])

 

    = 2(xy + 100/x + 100/y)

 

calculate ∂A/∂x and ∂A/∂y and set  each equal to 0.

 

Using those two equations solve for x and y (they won't be whole number so adjust your answer.)

 

Then solve for z to that will give xyz=100.

 

My guess is the dimensions to minimize the area of each box should be 4,5,5.

 

 

Any questions?