Question

A carton can hold 1'000 unit cubes that measure 1 inch by 1inch by 1 inch . How can describe the dimensions of the carton using unit cubes

Answer 1

You can't tell the dimensions from that information. 
All you can say is that the length multiplied by the width,
and then that product multiplied by the height, is 1,000 . 
But you can't say what the length or the width or the height is.

A few possibilities:

Length  Width  Height
      1            1        1000
      2            1          500
      4            5            50
      5            8            25
    10          10           10
    10          20             5  

There are many more possibilities.