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

You've told us the volume of the carton.

That doesn't tell us the dimensions.

If the unit cubes have to exactly fit the dimensions,

then all we know is that the dimensions are integers,

and their product is 1,000.

The carton could be

1 by 1 by 1,000

1 by 2 by 500

1 by 4 by 250

1 by 5 by 200

1 by 8 by 125

1 by 10 by 100

1 by 20 by 50

1 by 25 by 40

2 by 2 by 250

2 by 5 by 100

2 by 10 by 50

.

.

and several other possible sets.