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Intern  Joined: 19 Sep 2010
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The interior of a rectangular carton is designed by a certain  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 72% (02:29) correct 28% (02:47) wrong based on 1822 sessions

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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. $$\sqrt{x}$$

B. $$\sqrt{\frac{2x}{3}}$$

C. $$\sqrt{\frac{3x}{2}}$$

D. $$\frac{2}{3}*\sqrt{x}$$

E. $$\frac{3}{2}*\sqrt{x}$$

Originally posted by naaga on 25 Feb 2011, 07:24.
Last edited by Bunuel on 16 May 2017, 04:25, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Aug 2009
Posts: 8292
Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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26
17
sagnik242 wrote:
Bunuel wrote:
naaga 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. 3√x
B. 3√[(2x)/3]
C. 3√[(3x)/2]
D. (2/3) 3√x
E. (3/2) 3√x

Given: $$length:width:height=3k:2k:2k$$, for some positive number $$k$$. Also: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$.

confused how you got from : $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$. can you break this down further please?

Hi,
$$k=\sqrt{\frac{x}{12}}$$ ..
height is 2k as ratios are 3k:2k:2k
so $$2k=2\sqrt{\frac{x}{12}}$$..
=> $$2k=\sqrt{8}\sqrt{\frac{x}{12}}$$..
$$2k=\sqrt{\frac{8x}{12}}$$..
$$height=2k=\sqrt{\frac{2x}{3}}$$..
hope this is what you were looking for
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Math Expert V
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naaga 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. $$\sqrt{x}$$

B. $$\sqrt{\frac{2x}{3}}$$

C. $$\sqrt{\frac{3x}{2}}$$

D. $$\frac{2}{3}*\sqrt{x}$$

E. $$\frac{3}{2}*\sqrt{x}$$

Given: $$length:width:height=3k:2k:2k$$, for some positive number $$k$$.

Also: $$volume=x=3k*2k*2k$$;

$$x=12k^3$$;

$$k^3=\frac{x}{12}$$;

$$k=\sqrt{\frac{x}{12}}$$;

$$height=2k=\sqrt{\frac{2x}{3}}$$.

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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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3
3
Simple plug and play here

Choose the numbers given (3:2:2) = 12 for volume. Then, plug 12 into X in the answer choices to get 2.

cuberoot(2x/3) > cuberoot(2(12)/3) > cuberoot(24/3) > cuberoot (8) > 2

I immediately started with B, since it makes since (to find V, it'll be a cuberoot of something, with some division involved), and blamo it worked.
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GMAT 1: 760 Q50 V44 Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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2
Picking numbers:

If l = 3, w = 2 and h = 2, volume = 12.

Now, testing:

sq rt 12 = 2* sq rt 3 = approx 2*1.73 = approx 3.46
A) 3* sq rt 12 = approx 10.4 WRONG
B) 3 *sq rt 8 = approx 3 * 2.8 = 8.4 WRONG
C) 3 * sq rt 18 = approx 3 * 3*sq rt 2 = 3 * 4.2 = 12.6 WRONG
D) (2/3) * 3.46 = approx 2.3
E) (3/2) * 3.46 = approx 5.1

I would choose D, but it is not exact. Did I do anything wrong?
E)
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Hi Bunuel,

Could you suggest any way to master these questions? I seem to know how to deal with them, but make silly mistakes every single time.

Thanks!
Math Expert V
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Posts: 8292
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Viktoriaa wrote:
Hi Bunuel,

Could you suggest any way to master these questions? I seem to know how to deal with them, but make silly mistakes every single time.

Thanks!

Hi,
It will be important to know at what stage do you go wrong..
1)formula stage..
2)calculations..
3)difficulty with variables..

For example in this Qs..
things one should know.

1)formula for Volume of RECTANGULAR BOX..
2) converting ratio 3:2:2 to numeric values by multiplying each term by common variable..
3) What one has to be careful is to realize
a) it is not square root but 3rd root
b) height is 2 * variable ..

the other way to do is to take same value for common term in ratio. find the volume..
work backwards by substituting V in choices to get height.

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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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Bunuel wrote:
naaga 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. 3√x
B. 3√[(2x)/3]
C. 3√[(3x)/2]
D. (2/3) 3√x
E. (3/2) 3√x

Given: $$length:width:height=3k:2k:2k$$, for some positive number $$k$$. Also: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$.

confused how you got from : $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$. can you break this down further please?
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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

We are given that the ratio of length: width: height = 3 : 2 : 2 and we are also given that the volume of the rectangular solid is x. We can use n as the variable multiplier for our ratio, giving us:

length: width: height = 3n : 2n : 2n

Now we are ready to determine the height in terms of x. _________________

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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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How x=12k to the power 3 becomes k= Cube root x/12? chetan2u Bunuel please help. I understood the later part of the problem. Thanks.
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How x=12k to the power 3 becomes k= Cube root x/12? chetan2u Bunuel please help. I understood the later part of the problem. Thanks.

$$x=12k^3$$;

$$k^3=\frac{x}{12}$$;

Take the cube root: $$k=\sqrt{\frac{x}{12}}$$;
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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1
1
chetan2u wrote:
sagnik242 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. 3√x
B. 3√[(2x)/3]
C. 3√[(3x)/2]
D. (2/3) 3√x
E. (3/2) 3√x

Given: $$length:width:height=3k:2k:2k$$, for some positive number $$k$$. Also: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$.

confused how you got from : $$k=\sqrt{\frac{x}{12}}$$ --> $$height=2k=\sqrt{\frac{2x}{3}}$$. can you break this down further please?[/quote]

Hi,
$$k=\sqrt{\frac{x}{12}}$$ ..
height is 2k as ratios are 3k:2k:2k
so $$2k=2\sqrt{\frac{x}{12}}$$..
=> $$2k=\sqrt{8}\sqrt{\frac{x}{12}}$$..
$$2k=\sqrt{\frac{8x}{12}}$$..
$$height=2k=\sqrt{\frac{2x}{3}}$$..
hope this is what you were looking for[/quote]

How does 2k become: $$\sqrt{8}\sqrt{\frac{x}{12}}$$..
Where you get the 8 from?
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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Bunuel wrote:
naaga 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. $$\sqrt{x}$$

B. $$\sqrt{\frac{2x}{3}}$$

C. $$\sqrt{\frac{3x}{2}}$$

D. $$\frac{2}{3}*\sqrt{x}$$

E. $$\frac{3}{2}*\sqrt{x}$$

Given: $$length:width:height=3k:2k:2k$$, for some positive number $$k$$.

Also: $$volume=x=3k*2k*2k$$;

$$x=12k^3$$;

$$k^3=\frac{x}{12}$$;

$$k=\sqrt{\frac{x}{12}}$$;

$$height=2k=\sqrt{\frac{2x}{3}}$$.

Hi Bunuel

Do you mind clarifying the red part? Thank you very much
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Kudos to Jeff. The only expert who explained the last step and really the only step that made this problem challenging.
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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glt13 wrote:
Simple plug and play here

Choose the numbers given (3:2:2) = 12 for volume. Then, plug 12 into X in the answer choices to get 2.

cuberoot(2x/3) > cuberoot(2(12)/3) > cuberoot(24/3) > cuberoot (8) > 2

I immediately started with B, since it makes since (to find V, it'll be a cuberoot of something, with some division involved), and blamo it worked.

Just to note this works with only multiples of the ratios given.

E.g. here you used 3n:2n:2n
n= 1

n=2 3(2):2(2):2(2)
6:4:4 and volume of 96
plug 96 into the solutions to get 4
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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For anyone who still might be confused. Here is a simpler way and quite frankly how i go about problems involving ratios.

Given : l:b:h = 3n:2n:2n and vol=x
Asked to find length in terms of x.

From the start convert whatever you need to find to unitary. This will make your life much easier and also help you avoid silly mistakes wherein you forget to multiply your answer by some factor to get the right answer.
Getting back...

Now, l:b:h = (3/2)n : n : n

l*b*h = x
(3/2)n*n*n = x
i guess i don't need to tell you how to solve for n now.
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Re: The interior of a rectangular carton is designed by a certain  [#permalink]

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naaga 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. $$\sqrt{x}$$

B. $$\sqrt{\frac{2x}{3}}$$

C. $$\sqrt{\frac{3x}{2}}$$

D. $$\frac{2}{3}*\sqrt{x}$$

E. $$\frac{3}{2}*\sqrt{x}$$

Let's plug numbers in!

We know that Length*Width*Height = Volume (Volume = X in this question)
And that the ratio of Length:Width: Height is 3k:2k:2k, K being a constant.
If K = 1, the volume is 12 and the Height is 2. So from the answer choices, we need to find an answer choice when you plug in x =12, you get 2. Only B does so Re: The interior of a rectangular carton is designed by a certain   [#permalink] 26 Nov 2019, 20:08
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