Step 1 :The volume of a rectangular box is given by the product of its length, width, and height. In this case, the original volume is \(4*4*2 = 32\) cubic inches.
Step 2 :The problem states that each dimension of the box is increased by the same amount, say x, to yield a new box with volume four times the old. This means the new volume is \(4*32 = 128\) cubic inches.
Step 3 :We can set up the equation \((4+x)(4+x)(2+x) = 128\) to solve for x.
Step 4 :The solution to the equation gives three possible values for x. However, since we are dealing with dimensions of a box, we can discard the complex solutions and only consider the real solution.
Step 5 :Final Answer: The amount each dimension of the original box was increased to create the new box is approximately \(\boxed{1.80}\) inches.