if the areas of three adjacent faces of a rectangle are in the ratio 2:3:4 and its volume is 9000 cm^3 what is the dimensions of the rectangle.?

Suppose the dimensions of the rectangle are x cm * y cm * z cm

9000 = xyz

xy : yz : zx = 2 : 3 : 4
9000/z : 9000/x : 9000/y = 2 : 3 : 4
1/z : 1/x : 1/y = 2 : 3 : 4
z : x : y = 1/2 : 1/3 : 1/4 = 6 : 4 : 3
So, z = 6k , x = 4k , y = 3k , for some k

9000 = xyz = (4k)(3k)(6k) = 72 k^3
k^3 = 9000 / 72 = 125
k = 125^(1/3) = 5
So, z = 6k = 30 , x = 4k = 20 , y = 3k = 15
x cm * y cm * z cm = 20 cm * 15 cm * 30 cm

Ans: 20 cm * 15 cm * 30 cm