The interior of

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shahfahad: Senior | Next Rank: 100 Posts; Posts: 42; Joined: Fri Oct 30, 2015 9:01 am

The interior of

by shahfahad » Thu Nov 26, 2015 2:04 am

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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 rootâˆšx
(B) cube rootâˆš2x/3
(C) cube rootâˆš3x/2
(D) 2/3 cube rootâˆšx
(E) 3/2 cube rootâˆšx

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Source: Beat The GMAT — Problem Solving | Thu Nov 26, 2015 2:04 am

vishalwin: Master | Next Rank: 500 Posts; Posts: 274; Joined: Fri Sep 18, 2015 10:58 pm; Thanked: 12 times; Followed by:1 members; GMAT Score:530

by vishalwin » Thu Nov 26, 2015 12:04 pm

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)

Thanks & Regards
vishalwin
------------------------------------
GMAT Score - 530
I will BEAT the GMAT!

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pm; Location: East Bay all the way; Thanked: 625 times; Followed by:119 members; GMAT Score:780

by Matt@VeritasPrep » Thu Nov 26, 2015 11:58 pm

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.