The "volume" equation is

    x*(x+1)*(x^2+1) = 120   cubic feet.


One solution can be easily guessed:  x = 3.


Indeed,  3*(3+1)*(3^2+1) = 3*4*10 = 12*10 = 120.


Next, notice that the function x*(x+1)*(x^2+1)  is monotonically increased.


It means that x = 3  is the UNIQUE solution to the "volume" equation, and THERE IS NO other real solution.


ANSWER.  Under given conditions, the unique set of the dimensions of the tank is 3 ft, 4 ft, and 10 ft.