The surface area of the cuboid is `1400cm^2.`

Since the ratios of the edges are 2:3:4 then letting x represent the factor, the sides have length 2x, 3x, and 4x.

Therefore the surface area of the cuboid can be represented by:

`2(8x^2) + 2(6x^2) + 2(12x^2)`

`16x^2 + 12x^2 + 24x^2`

`52x^2` = `1400`

` ` `x^2 = 1400/52`

`x =sqrt(1400/52)`

`x =sqrt(350/13)`

Each side of ratio 2:3:4 is:`2sqrt(350/13)` , `3sqrt(350/13)` , and `4sqrt(350/13)`

Therfore to find the volume of the cuboid: length x width x height is:

`2sqrt(350/13)* 3sqrt(350/13)* 4sqrt(350/13)` = `2*3*4* sqrt(350/13)^(3)`

`12(350/13)*sqrt(350/13)`

`4200/13*sqrt(350/13)`

If I rationalize the denominator for `sqrt(350/13)`  I get `sqrt(4550)/13`

`4200/13*sqrt(4550)/13` = `(4200sqrt(4550))/169` `~~` 1676.364 `cm^3`

The volume of the cuboid is approximately 1676.364 cubic centimeters.