geometry

[earth@work]

The interior of a rectangular carton is designed by a certain manufacturer to have a volume of x cubic feet and a ratio of length to width to height of 3:2:2. In terms of x, which of the following equals the height of the carton, in feet?

A. ³√x
B. ³√[(2x)/3]
C. ³√[(3x)/2]
D. (2/3) ³√x
E. (3/2) ³√x

Ans: B

[DanaJ]

Since you have a ratio of length to width to height of 3:2:2, then we can safely assume that the size of the length is 3a and the size of width = size of height = 2a. This makes the volume of the box 3a*2a*2a = 12a^3 = x. This in turn makes a = third degree root of (x/12), thus making 2a = 2*[third degree root of x/12)]. What you need to do now is to "insert" the 2 in front of the third degree root "into" the third degree root and you get that 2a or the height is [third degree root of (8x/12)]. Simplify by 4 under the third degree root and you get height = third degree root of (2x/3).

Edit: I added a photo of the thing, since I hate writing in words third degree root...

[earth@work]

thanks!