Question

A rectangular safe is
to be made of steel of uniform thickness, including the door. The inside
dimensions are 1.20m, 1.20m and 2.00m. If the volume of steel is 1.25m^3



















, find its thickness 

Answer 1

This question I answered earlier.  I will like to solve in an easier method  which u can follow in such problems with cubic equations. It is by successive iteration.

The INSIDE dimensions of the rectangular safe:
        Length L = 1.20 m        Breadth B = 1.20 m          Height H = 2.00 m
        Inside volume = 1.2² * 2 = 2.88 m³
Let thickness of safe be t meters

To get the outside (external) dimensions: add t on each side of a dimension (left side and right side for width and length. and at top and bottom for height).  So each dimension becomes plus 2t.

Outside Length = 1.20+2t   External width  = 1.20+2t  external height = 2.00+2t
External (Outside) volume = (1.2+2t)² (2+2t)  = 2²(0.6+t)² 2 (1+t)

The volume of Steel used is the difference between the outside volume and the inside volume.

Hence :  8 (t+0.6)² (t+1) - 2.88 = 1.25 meter³

                (t+0.6)² (t+1) = 4.13 / 8 = 0.51625 
  
           Let (t + 0.6) = x
                     x² (x+0.4) = 0.51625             or x³ + 0.4 x² - 0.51625 = 0

We can try the ways of factorization and finding zeros. But here I show another way of successive iterations to solve. rewrite the equation as,

            x² = 0.51625 / (x+0.4)    or  x = √[0.51625 / (x+0.4)]
                  x  = 0.7185 / √(x+0.4)

We will find the value of x by starting with some value and calculating it many times, so that we get the precision we want.   Substitute a value of x on R H S and find the L H S

Start with x = 1 ,    new value of  x =  0.7185 / √(1+0.4) = 0.6072
       use x = 0.6072,  new value of x = 0.7185 / √(0.6072+0.4) = 0.7159
             next value of x in iteration = 0.7185 / √(0.7159+0.4) =0.68016
               Next iteration x = 0.6913 
                 next iteration x = 0.68778
                   next iteration x = 0.6889
                     next iteration x = 0.68854
                       next iteration x = 0.68866
                        next iteration x = 0.6886...

Let us take x = 0.6886  as these digits are repeated in the next iteration.
   t = x - 0.6 = 0.0886  meters

or    t = 8.86 cm   approximately

Verification of answer :

      Outside volume = (1.2+2*0.0886)² (2+2*0.0886) = 4.1294 m³
      Inside volume = 1.2² * 2  = 2.88 m³
          Volume of steel = 1.2494  m³   ≈ 1.25 m³ 

================================
 as  x³ + 0.4 x² - 0.51625 = 0
         x ³ = (0.51625 - 0.4 x²)
              x = ∛(0.51625 - 0.4 x²)      

we can use this equation also for successive iterations.