The interior of a rectangular carton is designed by a certain

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.



Answer: B

The interior of a rectangular carton has a ratio of length to width to height of 3:2:2.
Let 3k = the length of the carton
Let 2k = the width of the carton
Let 2k = the height of the carton

Note: Our goal is to find an equivalent value to 2k (the height of the carton)

Note: This guarantees that the ratio of length to width to height of 3:2:2

The carton's volume is x.
Volume of carton = (length)(width)(height)
So, we can write: (3k)(2k)(2k) = x
Simplify: \(12k^3 = x\)
Divide both sides by 12 to get: \(k^3 = \frac{x}{12}\)
Take the cube root of both sides to get: \(k = \sqrt[3]{\frac{x}{12}}\)
Multiply both sides of the equation by 2 to get: \(2k = 2\sqrt[3]{\frac{x}{12}}\)

Check the answer choices.....\(2\sqrt[3]{\frac{x}{12}}\) is not among them.
So it looks like we need to find an equivalent way to express \(2\sqrt[3]{\frac{x}{12}}\)
Since \(2= \sqrt[3]{8}\), we can write: \(2\sqrt[3]{\frac{x}{12}} = (\sqrt[3]{8})(\sqrt[3]{\frac{x}{12}})= \sqrt[3]{\frac{8x}{12}}= \sqrt[3]{\frac{2x}{3}}\)

Answer: B

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)

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.

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?

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.

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: \(k=\sqrt[3]{\frac{x}{12}}\);

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.
Straight B

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

Answer is B

Let the length, width, and height \(=3m, 2m, \ and \ 2m\)

The volume of the rectangle \(= 3m*2m*2m=12m^3\)

\(Given \ that, 12m^3=x\)

\(m^3=\frac{x}{12}\)

\(m=3\sqrt{\frac{x}{12}}\)

\(2m=2*3\sqrt{\frac{x}{12}}\)

\(2m=3\sqrt{\frac{x*8}{12}}\)

\(2m=3\sqrt{\frac{2x}{3}}\)

The answer is B

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:

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

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?

#Approach 1

Its given that:
length : width : height = 3 : 2: 2

Assume that length = \(3k\), width = \(2k\) , height = \(2k\)

Volume of a cuboid = Length * width * height = \(3k * 2k * 2k = 12 k^3 = x\)

\(k^3 = \frac{x}{12}\)

\(k = \sqrt[3]{\frac{x}{12}}\)

Height of the carton = \(2k = 2*\sqrt[3]{\frac{x}{12}} =\sqrt[3]{\frac{8x}{12}} = \sqrt[3]{\frac{2x}{3}}\)

Option B

#Approach 2
Assume that length = 3 feet , width = 2 feet, height = 2 feet.

Volume = 3*2*2 = 12 = x

So Substitute x = 12 in answer options and check whether you are getting h = 2.
Only option B will satisfy the above condition.

\(\sqrt[3]{\frac{2x}{3}}\) = \(\sqrt[3]{\frac{2*12}{3}}\) = \(\sqrt[3]{8} = 2\)

Option B

Thanks,
Clifin J Francis,
GMAT SME

Answer: Option B

Step-by-Step Video solution by GMATinsight



BEST On-Demand QUANT Course: Topicwise Modules | 2000+Qns | Official Qns. | 100% Video solution.
Two MUST join YouTube channels : GMATinsight (1000+ FREE Videos) and GMATclub

chetan2u
I am really struggling with the smart number strategy on quant. In this case, I followed the constraints and arrived at an area of 12 (3 times 2 times 2). I then plugged 12 into x to see what yields a height of 2. I saw that choice B gave me an answer of 2.
Do you have to test all of the answers (e.g., plug x=12 into the remaining answer choices to make sure they do not also yield 2)? How do I know if I need to choose another set of numbers that meets those constraints (e.g., 6:4:4 --> 6 times 4 times 4 =96, and then test the answers to see if Choice B gives me 4)? Thank you in advance!