Very confusing optimization question!

Question

I am really confused about this equation and how do you derive the volume and area equation!

” An open-top rectangular box is to be 6 meters high and have a volume of 300 cubic meter. Find the dimensions for which the amount of material needed to construct the box is minimum”

Answer 1

The volume and surface area are from geometry.  Let x = the length, y = the width, and h = the height of the box.  The height is given as 6 m and the volume is given as 300 m^3:

 

Volume = height x length x width

300 = 6xy

50 = xy

 

The surface area is:

 

SA = area of base + area of the four sides

SA = xy + 2xh + 2yh

SA = 50 + 12x + 12y     [xy=50 from the volume equation and h = 6]

 

You want to minimize SA.  First, use substitution from the volume equation to eliminate the y variable:

 

50 = xy

50/x = y

 

SA = 50 + 12x + 12(50/x)

SA = 50 + 12x + 300x^-1

 

Take the derivative of SA wrt x, set it to zero, and solve for x.  That is the x value that minimizes the SA.  The corresponding y value is 50/x.