Three identical cylindrical cans of radii 2 inches are to be shipped

Question

d1.JPG Three identical cylindrical cans of radii 2 inches are to be shipped in a rectangular box. The figure shows a cross-section with the distribution of the cans so that the minimum space in the container is wasted. What is, in square inches, the whole area of the cross-sections? A. 64 B. 12\pi C. (4 + 2 \sqrt3)^2 D. (16+2 \sqrt3) E. It cannot be determined with the information given. Could any one clarify what do they exactly mean by cross-section? I am not sure if I am getting it right ... Thanks!

Bunuel · Accepted Answer

d1.GIF The area of the rectangle is length*width . Length equals to 4*radius=8 . Width will be altitude of the equilateral triangle formed by the radii of the circles (shown on the diagram) plus 2*radius=4 . Altitude of the equilateral triangle is s\frac{\sqrt{3}}{2} (where s is the side of the equilateral triangle), in our case s equals to 2radius=4 --> altitude=4\frac{\sqrt{3}}{2}=2\sqrt{3} . So the width of the rectangle will be 4+2\sqrt{3} . Area=length*width=8*(4+2\sqrt{3})

esledge · Answer

Since "the figure shows a cross section," I would think the "whole area of the cross section" is the area of the overall figure, i.e. the rectangle.

But none of the answers listed is correct. I get 8(4 + 2  \sqrt{3}  ).

Nusa84 · Answer

Ok, I get it, its weird that they put the answer wrong ... So, just to confirm, we understand the "whole area of the cross section" means the whole figure (rectangle) right?

esledge · Answer

Yes. The wording here strikes me as odd and, combined with the mistake in the answers, makes me think this question is either not from a good source or has been transcribed incorrectly as it has been passed around.

The actual GMAT would more likely say "What is the area of the cross section of the rectangular box in square inches?" to be clear. (See OG PS #145 for an example.)

Nusa84 · Answer

Thanks for the clarification!

I dont really know the exact source as i get it from a combination that my gmat school prepared. But i think the answer is correct tough, if you factorize 2 in the terms of the parenthesis you get the 16(2+sqrt3) marked as a correct answer.

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nonameee · Answer

I don't understand the above sentence.

Well, actually reading the whole problem again, I must say I don't quite understand at all? I thought we are asked to find the area of a rectangular which I thought should be a square (4*r = 8). So, I thought the answer would be 8*8 = 64. Then I have read your explanation and now I don't know what the question asks.

So maybe if you could clarify the whole question.

Thanks a lot.

Bunuel · Answer

We should find the area of the box. Note that the box is not a square it's just a rectangle, so the length of it is indeed 4r=8, but the width is less. If you look at the diagram you'll see that it equals to 2r (one radius below the triangle and one radius above the triangle) plus the height of the triangle (shown at the diagram): 2r+height. so the area=8r*(2r+height).

Hope it's clear.

nonameee · Answer

Oh, yes, thank you, Bunuel. I got it now.

KarishmaB · Answer

First instinct - It looks like a square. But remember - "Figures may not be to scale. Don't assume anything from figures. Check it out for yourself" When would it be a square? Ques.jpg But the circles have been placed to maximize area utilization. So the width of this rectangle is actually less than its length. Ques1.jpg We know the length here is still 8. What we need is the width AC. Now AB = 2 because it is equal to the radius Also OC = 2, again equal to radius. But what is OB? Note that the triangle in the middle is equilateral because each of its sides is 2r i.e. 4. Hence the altitude of this triangle, i.e. OB, will be {\sqrt{3}*4}/2 (Altitude of equilateral triangle of side a is {\sqrt{3}*a}/2 ) therefore AC becomes 2 + \sqrt{3}*4/2 + 2 = 4 + 2\sqrt{3} Area of rectangle = 8* (4 + 2\sqrt{3}) or 16* (2 + \sqrt{3})

Sarang · Answer

The 3 midpoints of the circles, when joined form an equilateral triangle with side- 4
now one side of a square= 2 + 2 + height of equilateral triangle

Height of equilateral triangle = side * sqrt(3)/2 = 4 * sqrt(3)/2 = 2sqrt(3)

total length of rectangle = 2sqrt(3)
total width of rectangle = 8
area= length * width

Answer:- choice not available....

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Three identical cylindrical cans of radii 2 inches are to be shipped

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