Problem Solving

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) cube root√x
(B) cube root√2x/3
(C) cube root√3x/2
(D) 2/3 cube root√x
(E) 3/2 cube root√x

OA: B


Let

L= 3k
W= 2k
H= 2k


Volume of rectangular solid i.e. Cuboid is = L*B*H = 12*(K^3) = x


so, k = cube root(x)/ cube root(12)


cube root of 12 = cube root (2*2*3)

cube root (12) = 2^(2/3) * 3^(1/3)

Height = 2k = 2 *{cube root(x)/ cube root(12)} = 2*cube root(x)/cube root(12)


multiply the number with 2^(1/3) (we are doing this because 2^(1/3) * 2^(2/3) in denominator will become 2 and will cancel out the 2 in numerator.)

hence, we will have cube root(x)* cube root(2)/cube root(3) = cube root(2x/3)

cube root (12) = 2^(2/3) * 3^(1/3)

Let's say our height = h.

We have

height = h
width = h
length = 1.5h

So ...

h * h * 1.5h = x

or

h = cube root of (x/1.5), or cube root of (2/3)x

Answer B is not formatted correctly, but it seems to be the intent of the problem.

Thanks