The inside dimensions of a rectangular wooden box are 4 mete

Question

The inside dimensions of a rectangular wooden box are 4 meters by 5 meters by 6 meters. A cylindrical drum is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such drums that could be used, what is the volume, in cubic meters, of the one that has maximum volume?

(A) \(20 \pi\)

(B) \(24 \pi\)

(C) \(25 \pi\)

(D) \(96 \pi\)

(E) \(100 \pi\)

(A)  20 \pi

(B)  24 \pi

(C)  25 \pi

(D)  96 \pi

(E)  100 \pi

harsh6239 · Accepted Answer

You have three options here. My approach is to calculate volumes for all of them and see which one is greatest.

1. Cylinder's base rests on a 4x5 side of the box. So, height becomes 6 and the maximum possible radius becomes 4/2.

So, maximum volume =  \pi . (4/2)^2 . 6 = \pi 24

2. Cylinder's base rests on a 5x6 side of the box. So, height becomes 4 and the maximum possible radius becomes 5/2.

So, maximum volume =  \pi . (5/2)^2 . 4= \pi 25

3. Cylinder's base rests on a 4x6 side of the box. So, height becomes 5 and the maximum possible radius becomes 4/2.

So, maximum volume =  \pi . (4/2)^2 . 5 = \pi 20

Option 2 gives maximum value and so that should be the answer, \pi 25.

In hindsight you can see that since in volume of a cylinder  (\pi r^2 h)  the radius r has a higher power, the one with a higher radius will have maximum volume when other parameters are close to each other (like 4,5 and 6). You get maximum radius when base is kept on a 6x5 face. Still, personally I will go with finding all three volumes and then comparing them.

Bunuel · Answer

"Stolen" question from GMAT Prep: Discussed here: the-inside-dimensions-of-a-rectangular-wooden-box-are-128053.html Similar question to practice: a-rectangular-box-has-dimensions-12-10-8-inches-what-is-the-28790.html Hope it helps.

scholar1 · Answer

I am a bid confused by this question.. so does the base of the cylinder have to be a circle and cannot be an oval shape?

EMPOWERgmatRichC · Answer

Hi scholar1,

By definition, a cylinder has a base that is a circle; this impacts a variety of questions that can be asked about a cylinder (volume, surface area, width, etc.). In this question, the width of the cylinder (re: the diameter) will be influenced by how it is "oriented" inside the box (for that matter, so will the height). Depending on which side is the "base" of the box, the maximum possible diameter of the cylinder (and the maximum height of the cylinder) will change.

GMAT assassins aren't born, they're made,
Rich

TestPrepUnlimited · Answer

We have three options:

(1) Use 4*5 as the base and 6 as the height
(2) Use 4*6 as the base and 5 as the height
(3) Use 5*6 as the base and 4 as the height.

However, the volume from (2) is strictly less than that of (1) so we only need to consider (1) and (3).

From (1) we can only use 4*4 of the base, so the radius is 2 and the volume is  2*2*\pi * 6 = 24\pi .

From (3) we can only use 5*5 of the base, so the radius is 2.5 and the volume is  2.5*2.5*\pi*4 = 25\pi .

Therefore (3) is bigger than we choose C.

Ans: C

wishmasterdj · Answer

Volume is maximised when you take min of the 2 highest values as the base, as the diameter of the drum (5 and 6, and thus 5 as diameter)

therefore pi * (5/2)^2 * 4 = 25 pi

ohmphanurut · Answer

I always wondered when facing this type of question. Why can't we take the max value as a diameter -> radius (6/2 = 3)? This will clearly maximize the volume in my opinion.

EMPOWERgmatRichC · Answer

Hi ohmphanurut, In this prompt, the dimensions that we are dealing with are 4x5x6, so the 'base' of the box is either 4x5 OR 4x6 OR 5x6. A diameter of 6 is NOT possible in any of these situations (the diameter would either be 4 or 5 depending on which side is the base). By extension, the height will also differ depending on the orientation of the box (the height will either be 4 or 5 or 6). To maximize the volume of the cylinder, you have to consider which side is the "base" of the box, since the possible diameter of the cylinder (and the height of the cylinder) can be three possible options. GMAT assassins aren't born, they're made, Rich Contact Rich at: Rich.C@empowergmat.com www.empowergmat.com

The inside dimensions of a rectangular wooden box are 4 mete

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