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# The interior of a rectangular carton is designed by a certain

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

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Updated on: 16 May 2017, 04:25
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32
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Difficulty:

55% (hard)

Question Stats:

70% (02:09) correct 30% (02:25) wrong based on 1038 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. $$\sqrt[3]{x}$$

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

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

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

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

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

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25 Feb 2011, 07:49
1
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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The interior of a rectangular carton is designed by a certain [#permalink]

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25 Feb 2011, 08:04
13
9
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. $$\sqrt[3]{x}$$

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

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

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

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

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

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

$$x=12k^3$$;

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

$$k=\sqrt[3]{\frac{x}{12}}$$;

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

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

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13 Mar 2016, 04:46
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!
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Posts: 5928
Re: The interior of a rectangular carton is designed by a certain [#permalink]

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13 Mar 2016, 05:00
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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16 Mar 2016, 15:57
1
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: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$.

confused how you got from : $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$. can you break this down further please?
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Posts: 5928
Re: The interior of a rectangular carton is designed by a certain [#permalink]

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16 Mar 2016, 19:45
13
8
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: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$.

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

Hi,
$$k=\sqrt[3]{\frac{x}{12}}$$ ..
height is 2k as ratios are 3k:2k:2k
so $$2k=2\sqrt[3]{\frac{x}{12}}$$..
=> $$2k=\sqrt[3]{8}\sqrt[3]{\frac{x}{12}}$$..
$$2k=\sqrt[3]{\frac{8x}{12}}$$..
$$height=2k=\sqrt[3]{\frac{2x}{3}}$$..
hope this is what you were looking for
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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04 May 2016, 21:45
1
2
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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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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05 May 2016, 04:59
3
3
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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17 Mar 2018, 00:31
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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The interior of a rectangular carton is designed by a certain [#permalink]

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17 Mar 2018, 03:18
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: $$k=\sqrt[3]{\frac{x}{12}}$$;
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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05 May 2018, 06:42
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: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$.

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

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

How does 2k become: $$\sqrt[3]{8}\sqrt[3]{\frac{x}{12}}$$..
Where you get the 8 from?
Re: The interior of a rectangular carton is designed by a certain   [#permalink] 05 May 2018, 06:42
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