An open-empty rectangular box with negligible wall thickness is 6 feet

Question

box.PNG An open-empty rectangular box with negligible wall thickness is 6 feet wide, 6 feet long, and 2 feet high. An ant starts crawling from one corner (point A) to reach the diametrically opposite corner (point B). If the ant takes the shortest route possible to reach the destination by crawling on the inner surface of the box, how many feet does the ant travel? A. 6\sqrt{2} B. \sqrt{10} C. 2\sqrt{19} D. 10 E. 14

ShukhratJon · Accepted Answer

Hola amigos

What would I do if I had this box and ant in real life? In order to show this little buddy the shortest path from A to B , I would definitely UNFOLD the box. Now the shortest route from A to B is the hypotenuse of the right triangle ABC. Thus AB = 10 .

Zhenek · Answer

If I understood the question correctly I really wonder if thats actually a gmat question. Not only my answer is different from what is proposed (my answer is 9 feet or approximately 9.54 feet) but also the way to solve it required me to use things which are most likely beyond the bounadires of gmat requirements.
Basically the way I approached this task was evaluating 2 scenarios:
 https://i.imgur.com/AWv6VXQ.png

and

https://i.imgur.com/jMojB5m.png

The first one gave me the lowest answer after conducting analysis on the distance function (sum of lengths of AC and CB)

Bunuel · Answer

Similar questions to practice: an-ant-is-clinging-to-one-corner-of-a-box-in-the-shape-of-a-135055.html s97-184708.html an-ant-crawls-from-one-corner-of-a-room-to-the-diagonally-134454.html Hope it helps.

itzmyzone911 · Answer

How did you arrive at your ans?? Explain and then we shall see whether this is within or beyond GMAT scope

itzmyzone911 · Answer

Bunuel you rock :-D

...you seem to have virtually every GMAT problem on earth on your finger tips..

Zhenek · Answer

Haha, I'm glad I actually looked at it again, anyway, I made a mistake in my calculations and it is indeed 10 in the end, damn. I'll just explain my way of thinking then:
My 2nd picture, the point C is on the line and X is a variable. Length function is  \sqrt{36+x^2} + \sqrt{36+(2-x)^2} .
With the help of derivatives I found that its lowest value is  2*\sqrt{37}  when X = 1 (although one could also apply some common sense there since if u try drawing the thing abit differently - as a double-rectangular figure omitting the folding, the shortest distance ends up being a straight line and it happens that since the rectangles have equal lengths, the line will cross the folding in its middle thus X = 1).  2*\sqrt{37}  > 12.
First picture is similar. Length function is  F(x) = \sqrt{x^2+4} + \sqrt{36+(6-x)^2} .
In order to find minimum value I went with the rule that this  X_0  is pretty much a root of an equation  F'(X_0) = 0 
 \frac{2*x}{\sqrt{x^2+4}} - \frac{2*(6-x)}{\sqrt{36+(6-x)^2}} = 0 
 x* \sqrt{36+(6-x)^2} = (6-x)* \sqrt{4+x^2} 
solving this you get x = 1,5 as the only fitting answer, y ends up being 10 which is the answer to the question

itzmyzone911 · Answer

Well dear, I appreciate your insights...very thorough indeed.

However, the question can be solved using basics of Pythagoras theorem...do visit the link suggested by Bunuel at

an-ant-crawls-from-one-corner-of-a-room-to-the-diagonally-134454.html

Relevant discussions have taken place along with diagrams to make the point better. Do revert if any doubts..

Zhenek · Answer

Yea I did understand the approach displayed there and I find it very clever. I guess I will elaborate with the changed picture:
 https://i.imgur.com/EHY3HrB.png 
AB' is the shortest path the ant has to take and its the straight line, thus we got a triangle ADB' where AB' = hypotenuse and is easily found to be equal to  \sqrt{36+64} = 10

chanakya84 · Answer

Hi-
I think there is a typo in the question. I believe either width or length needs to be 2, if the answer is indeed the correct answer.
Using the data given the question:
Basically if we unfold the box, this will give us two rectangles, 1 rectangle with height 2 and base 6, 2nd rectangle with height 2 and base 6. The two rectangles have point A and B as diametrically opposite points. Now to find AB, it's sqrt( (6+6)^2 + 2^2) and that's not the correct answer.
 Bunuel  - I have followed the other posts that you have mentioned, but not sure what I am missing here in this question. Any inputs will be very helpful.

Thanks,
Chanakya

bumpbot · Answer

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An open-empty rectangular box with negligible wall thickness is 6 feet

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An open-empty rectangular box with negligible wall thickness is 6 feet

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