Unable to proceed after calculating x. Help.

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aman88: Master | Next Rank: 500 Posts; Posts: 142; Joined: Sat Oct 20, 2012 3:31 amFollowed by: 3 members; Location: New Delhi; Thanked: 20 times

by aman88 » Fri Dec 14, 2012 7:47 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 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

3y*2y*2y = x (given)
12y^3 = x
y = cube root(x/12)

Inserting this value of y in 2y...
H = 2y
H = 2*{cube root(x/12)}

I am stuck after this. How do I proceed?

Thanks.

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Source: Beat The GMAT — Problem Solving | Fri Dec 14, 2012 7:47 am

Anindya Madhudor: Senior | Next Rank: 100 Posts; Posts: 85; Joined: Mon Nov 12, 2012 11:12 amFollowed by: 2 members; Thanked: 8 times

by Anindya Madhudor » Fri Dec 14, 2012 7:53 am

Take 2 inside the cube root. Then,

H = cube root(8x/12) = cube root(2x/3)

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by GMATGuruNY » Fri Dec 14, 2012 9:04 am

aman88 wrote: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=3, W=2 and H=2.
x = volume = 3*2*2 = 12.
The question stem asks for the value of H=2. This is our target.
Now we plug x=12 into the answers to see which yields our target of 2.
A quick scan of the answer choices reveals that only B works:
(2/3 * x )^(1/3) = (2/3 * 12)^(1/3) = 8^(1/3) = 2.

The correct answer is B.

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aman88: Master | Next Rank: 500 Posts; Posts: 142; Joined: Sat Oct 20, 2012 3:31 amFollowed by: 3 members; Location: New Delhi; Thanked: 20 times

by aman88 » Fri Dec 14, 2012 10:40 am

Anindya Madhudor wrote:Take 2 inside the cube root. Then,

H = cube root(8x/12) = cube root(2x/3)

Anindya, can you please explain this? How did we take "2 inside the cube root"?

Thanks.

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Anindya Madhudor: Senior | Next Rank: 100 Posts; Posts: 85; Joined: Mon Nov 12, 2012 11:12 amFollowed by: 2 members; Thanked: 8 times

by Anindya Madhudor » Fri Dec 14, 2012 10:45 am

2 is equal to cube root of 8.

So, the expression "cube root(8x/12)" is equal to "2 * cube root(x/12)". Think of this in the reverse way. If you are given "cube root(8x/12)", can you write this as "2 * cube root(x/12)"? The answer is yes, you can. So, the opposite is true as well.

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ritind: Senior | Next Rank: 100 Posts; Posts: 47; Joined: Wed Nov 21, 2012 10:40 pm; Thanked: 4 times

by ritind » Wed Dec 19, 2012 2:39 am

Ver simple approach
Lets assume that the following are the dimensions of the rectangle
Length = 3, Width = 2, Height = 2
These figures comply with the ratio given in the ques.
Vol = 3*2*2 = 12

Cube root of 12 gives decimal value whereas the height that we have taken is an integer.
This eliminates A, D and E
Insert the value in B

cube root((2*12)/3) = 2 and voila B is the option

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Re: Unable to proceed after calculating x. Help.

by [email protected] » Thu Apr 22, 2021 2:35 pm

Hi All,

We’re told that the interior of a rectangular carton holds a volume of X cubic feet and the ratio of Length : Width : Height = 3 : 2 : 2. We’re asked which of the 5 answers is equal to the HEIGHT in feet. While the answers to this question “look scary”, we can answer this question rather handily by TESTing VALUES.

IF… Length = 3, Width = 2 and Height = 2, then the Volume = X = (3)(2)(2) = 12 cubic feet. This means that we’re looking for an answer that equals 2 when X = 12

Since the cube-root of 8 = 2 and the cube-root of 27 = 3, we know that the cube-root of 12 will NOT be an integer (and will likely be an ugly decimal). Since we’re looking for an answer that equals 2 (re: a nice, round integer), we can quickly eliminate several answers because they will clearly NOT be integers. Eliminate Answers A, D and E

Between the two answers that remain, since both involve cube-roots and we need the final result to equal 2, we’re looking for a total under the radical that equals 8. There’s only one answer that matches…

Final Answer: B

GMAT Assassins aren’t born, they’re made,
Rich

Contact Rich at [email protected]

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