Lesson Minimize surface area of a rectangular box with the given volume

Minimize surface area of a rectangular box with the given volume

Problem 1

Let x, y and z be the dimensions of a rectangular prism.


Then its volume is  

    xyz = 250 cm^3,     (1)


a fixed value, and the problem wants we minimize the surface area  2xy + 2yz + 2xz  under this restriction (1).


It is the same as to minimize the function of 3 independent variables

    F(x,y,z) = xy + yz + xz

under restriction (1).


Using the restriction, we can reduce function F(x,y,z) to function of two independent variables

    f(x,y) = xy +  + 

and look for the minimum of this function.


To find its minimum, we take partial derivatives of  f(x,y)  over x and y  and equate them to zero.
It gives us this system of equations

    y -  = 0,    (2)     (x-derivative)
  
    x -  = 0.    (3)     (y-derivative)


From (2) and (3)

    x^2*y = 250    (4)

    x*y^2 = 250.   (5)


Dividing (4) by (5), we get

     = 1,  or  x = y.


Working similarly with the other pair of independent variables, we can get similarly  x = z,

which tells us that the minimum surface value is achieved for the cube x = y = z.


Its dimension is   = 250,  or  x =  = 6.3  (rounded to one decimal place).


ANSWER.  The minimum surface area is a cube with the edge size of   = 6.3 cm.

CHECK.   = 250.047 cm^3.