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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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25 Feb 2011, 06:24
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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. $$\frac{3}{2}*\sqrt[3]{x}$$
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 May 2017, 03:25, 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, 06:49
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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, 07:04
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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. $$\sqrt[3]{x}$$

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: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$.

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

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25 Feb 2011, 08:36
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Ouch... I didn't understand the symbols.
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Posts: 16
Re: The interior of a rectangular carton is designed by a certain [#permalink]

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13 Mar 2016, 03:46
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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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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13 Mar 2016, 04:00
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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13 Mar 2016, 06:59
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.
[/quote]

Hi,

Thanks a lot for such a detailed reply! I think I just go wrong with recognizing equations themselves - I mean, in this case, I new formula of the volume and paid attention to some tricky parts, but didn't use variable in 3:2:2... Probably just need more attention....
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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16 Mar 2016, 14:57
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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: $$volume=x=3k*2k*2k$$ --> $$x=12k^3$$ --> $$k=\sqrt[3]{\frac{x}{12}}$$ --> $$height=2k=\sqrt[3]{\frac{2x}{3}}$$.

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

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16 Mar 2016, 18:45
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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, 20:45
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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, 03:59
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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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05 May 2016, 09:37
volume = lbh
2k.3k.2k = x

k = cuberoot of x/12

2 times cuberoot of x/12 = 3√[(2x)/3]
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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21 Jul 2017, 17:05
A (bit) different and easier way to do this

Phase 1

Ratios $$L:W:H = 3:2:2 = 3/2:1:1$$

So $$3/2*H = L$$
Also $$W = H$$

Phase 2

$$Volume = L*W*H = (3/2*H)H*H = 2/3*H^3 = x$$

Solve for H..

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

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03 Sep 2017, 23:31
Ans is B

L:B:H = 3:2:2
volume is LBH = x
3y.2y.2y = x
12y^3=x
y^3= x/12
2y = 2(x/12)^(1/3)
H= (8x/12)^(1/3)
= cuberoot(2x/3)
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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09 Sep 2017, 21:40
Can someone tell me what is wrong with my logic here:

LWH = X

width equals height so

LH^2 = X

3L = 2H so

2/3H*H^2 = X

2/3H^3 = X

H = (3/2X)^1/3

Thank you if you can point me in the right direction here.
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The interior of a rectangular carton is designed by a certain [#permalink]

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09 Sep 2017, 23:47
L:W:H IS IN RATIO OF 3:2:2
WHICH DOES NOT MEAN THAT 3L =2H
AND IF YOU THINK SO ThisandThat
THEN YOU NEED REVISION OF RATIOS SERIOUSLY
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Last edited by sahilvijay on 10 Sep 2017, 21:25, edited 1 time in total.
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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10 Sep 2017, 14:09
Why would L = 3/2H ?

If 3L : 2H then L = 2/3H correct?
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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10 Sep 2017, 19:54
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Since u r not applying ratio and proportion
Take the values 3 and 2 and check yourself

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

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10 Sep 2017, 21:25
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ThisandThat wrote:
Why would L = 3/2H ?

If 3L : 2H then L = 2/3H correct?

I CAN NOT UNDERSTAND
3L :2H DOES NOT MEANS L=2/3 H

SEE THIS

3L:2H => IT DOES NOT MEANS 3L = 2H => RATHER L:H = 3/2 => I GUESS YOU ARE SERIOUSLY CONFUSED AND NEED REVISION OF RATIOS AND PROPORTIONS
I CAN SEE WHY ARE YOU CONFUSED

YOU TELL ME IF A:B IS 6:4
IT MEANS
A/B = 6 /4 = 3/2
A:B = 3:2
MEANS 2 TIMES OF A = 3 TIMES OF B

I HOPE THIS MAKES SENSE AND IF STILL DOES NOT=> LET ME KNOW => ALTHOUGH YOU REQUIRE REVISION OF RATIO AND PROPORTION IS WHAT I CAN SEE.

I HAVE TRIED TO EXPLAIN
NO WHERE IT IS WRITTEN IN QUES THAT 3L=2H => SHOW ME IF YOU CAN??
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Re: The interior of a rectangular carton is designed by a certain [#permalink]

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10 Sep 2017, 22:45
I see now. Thank you very much that makes sense.
Re: The interior of a rectangular carton is designed by a certain   [#permalink] 10 Sep 2017, 22:45
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