The Interior of a Rectangular Carton is Designed by a Certain Manufacturer GMAT Problem Solving

Question - 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?

1. \(^3\sqrt{x}\)
2. \(^3\sqrt{\frac{2x}{3}}\)
3. \(^3\sqrt{\frac{3x}{2}}\)
4. \(\frac{2}{3}* {^3\sqrt{x}}\)
5. \(\frac{3}{2}* {^3\sqrt{x}}\)

‘The interior of a rectangular carton is designed by a certain manufacturer to have a volume of x cubic feet' - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1

Explanation:

It is asked which of the following equals the height of the carton, in feet. In terms of x, if the interior of a rectangular carton is designed by a certain manufacturer to have measurements:

• volume of x cubic feet
• a ratio of length to width to height of 3:2:2

Let 3m, 2m, and 2m be the length, width, and height.

The rectangle's volume is 3m2m2m.= 12m3

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

\(m3=\frac{x}{12}\)

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

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

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

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

The answer is B which is \(^3\sqrt{\frac{2x}{3}}\)

Correct Answer: B

Approach Solution 2

There is another approach to answering this question.

It is asked which of the following equals the height of the carton, in feet. In terms of x, if the interior of a rectangular carton is designed by a certain manufacturer to have measurements:

• volume of x cubic feet
• a ratio of length to width to height of 3:2:2

Here is a more straightforward approach, which is used in ratio-related problems.

It is Given : l:b:h = 3n:2n:2n and vol=x.
To calculate length in terms of x:

Convert whatever is needed to find it to unitary right away. This can greatly simplify our process and prevent us from making simple mistakes. Mistakes like forgetting to multiply our answer by a certain number to obtain the correct answer.

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

l*b*h = x
(3/2)n*n*n = x
\((3/2)n^3\) = x
n = \(^3\sqrt{\frac{2x}{3}}\)

The answer is B which is \(^3\sqrt{\frac{2x}{3}}\)

Correct Answer: B

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