Let x = the length of each side of the middle cube.
Without using the cube root function on the calculator, we shall iteratively find the value of x as follows:
We know that the volume is 30, and it lies between 27 and 64. Therefore x lies between 3 and 4.
Create a table as shown below.
We shall guess the value of x between iterations and calculate the volume as V = x*x*x.
When the absolute value of the difference between 30 and V is less than 0.5, we shall assume that the value of x is sufficiently accurate.
To make the next guess for x, we decrease x when V>30 and increase x when V<30.
Iteration x V=x*x*x |V-30| Decision
------------ ------------- ------------------- ------------ -------------
1 3.5 42.875 12.875 Lower x
2 3.25 34.328 4.328 Lower x
3 3.125 30.518 0.518 Lower x
4 3.0625 28.723 1.277 Raise x
5 3.0938 29.613 0.387 Good enough
Answer: The approximate length of the side is 3.0938.