M05-18

Question

To build a rectangular chicken pen, Mike has 40 meters of netting. If Mike wants to maximize the area of the pen, what will be the most favorable dimensions?

A. 12 x 8
B. 15 x 8
C. 10 x 10
D. 15 x 15
E. 15 x 5

Bunuel · Accepted Answer

Official Solution:

To build a rectangular chicken pen, Mike has 40 meters of netting. If Mike wants to maximize the area of the pen, what will be the most favorable dimensions?

A. 12 x 8
B. 15 x 8
C. 10 x 10
D. 15 x 15
E. 15 x 5

Approach 1: 
 
Let's denote the length and width of the rectangle as  x  and  y  respectively.
 
The perimeter  =2x+2y=40 . By simplifying this equation, we get  x+y=20 . Our aim is to maximize the area, which is the value of  xy .
 
It's a useful property to remember that:  for a given sum of two numbers, their product is maximized when they are equal . Thus, the value of  xy  will be maximized when  x=y=10 .
 
 Approach 2: 
 
This problem can be easily tackled if one is aware of the following property:  a square has a larger area than any other quadrilateral with the same perimeter .
 
Thus, to maximize the area, our rectangle should actually be a square. This gives us the equation for the perimeter as  4x=40 . Solving this equation gives  x=10 .

Answer: C

happyface101 · Answer

Isn't this problem flawed? The prompt specifically asks for a RECTANGULAR pen, so I excluded C and D b/c they are SQUARE pens??

Edit: OK, so Square is a special kind of Rectangle. Keeping this post as this is an important subtle point here for this problem.

Bunuel · Answer

Yes, every square is a rectangle but not vise-versa.

BoomHH · Answer

Hi  Bunuel ,

it's may a stupid question, but how could you build a rectangle with the dimension 15x15 if you have only 40m netting ?

Thanks

generis · Answer

IMO, not a dumb question. Knowing why answers are wrong and what kind of wrong answers to expect helps in similar problems to avoid such answers on similar problems.

My guess is that "15x15" is designed to hijack the person who, per Bunuel 's second explanation, correctly deploys "max area is square" but then, without actually thinking about the arithmetic, incorrectly decides that if a square is good, a bigger square is better.

TheNightKing · Answer

Either you know for a fact that area of a rectangle is maximum when it is a square which brings you to each side as 10. Area=100.
Or
You can check options. Hopefully, you boil down to C and D.
In case of C: 15*15 means other two sides are 5*5. Area=length*breadth=15*5=75.
In case of D: 10*10 means other two sides are 10*10. Area=length*breadth=10*10=100.

Thank you!

AndrewN · Answer

TheNightKing  - I am not sure I agree with the statement, "Hopefully, you boil down to C and D." I understand the logic behind thinking that the other sides would be 5 and 5, but  a 15 x 15 area indicates a square already , one with an area of 225. That is, you cannot remove one number or the other from the area and replace it with a reasonable alternative. What you have outlined is a 15 x 5, the same as choice (E), which is  not  the same thing as the 15 x 15 of choice (D). Answers (B) and (D) fail because such dimensions would not provide a suitable perimeter for the amount of netting Mike has at his disposal: choice (B) would give 2(15 + 8), or 46 meters, and choice (D) would yield 2(15 + 15), or 60 meters. Thus, only choices (A), (C), and (E) represent valid areas within the given information. At that point, it should not take long to figure out that (C) will be the answer, 100 compared to 96 (choice (A)) or 75 (choice (E)), with or without an understanding that squares maximize the area of  rectangular  polygons. (It would really maximize the area if Mike were building a circular enclosure, but that would be a different question altogether.)

I am not writing my response as a stern lecture, but I want any readers who come across your analysis to understand an important distinction. Thank you for sharing your thoughts.

- Andrew

TheNightKing · Answer

TheNightKing  - I am not sure I agree with the statement, "Hopefully, you boil down to C and D." I understand the logic behind thinking that the other sides would be 5 and 5, but  a 15 x 15 area indicates a square already , one with an area of 225. That is, you cannot remove one number or the other from the area and replace it with a reasonable alternative. What you have outlined is a 15 x 5, the same as choice (E), which is  not  the same thing as the 15 x 15 of choice (D). Answers (B) and (D) fail because such dimensions would not provide a suitable perimeter for the amount of netting Mike has at his disposal: choice (B) would give 2(15 + 8), or 46 meters, and choice (D) would yield 2(15 + 15), or 60 meters. Thus, only choices (A), (C), and (E) represent valid areas within the given information. At that point, it should not take long to figure out that (C) will be the answer, 100 compared to 96 (choice (A)) or 75 (choice (E)), with or without an understanding that squares maximize the area of  rectangular  polygons. (It would really maximize the area if Mike were building a circular enclosure, but that would be a different question altogether.)

I am not writing my response as a stern lecture, but I want any readers who come across your analysis to understand an important distinction. Thank you for sharing your thoughts.

- Andrew

Hi Andrew, Thanks for your response.
But I am confused.
but  a 15 x 15 area indicates a square already , one with an area of 225.  How? Then the perimeter becomes 2(15+15)=60. And that is not right. 
 Also I do understand that option D and E are same as per my logic and that is not the case but then what does 15*15 represent and what will be the other 2 sides.

From the point of view of the answer to the question, I have no issues since I know the logic of square having the maximum area but I guess I just confused myself.

Please help!

AndrewN · Answer

TheNightKing  - Per the part in blue, yes, that is  not  right. A perimeter of 60 would be problematic indeed, since Mike has only 40m of netting. To help with the second part, a 15 x 15 rectangle indicates a  length  of 15 and a  width  of 15, and since the length and width will not change from one parallel side of a rectangle to the other, there is no "extra" 5 and 5 to consider. I drew out a couple figures on Google Docs, which I am uploading here as a PDF, to illustrate what I mean. The first figure could correctly be called a 15 x 15, the second a 15 x 5. The third is a sketch, really, in which two segments that are each 15m in length are joined in some manner that is different from the second figure. But this begs the question, how  could  two 5m segments be joined to the figure in any meaningful way to produce a four-sided polygon that could be called a  rectangle ? That is the point I was hoping to make.

I like to say that as long as you get the correct answer (without cheating), how you got there does not matter--the GMAT™ does not award bonus points for elegant solutions. On the other hand, when you want to guide others, I think your reasoning has to be airtight, or else your explanations can be misleading to those who might take them at face value.

Thanks for the response. I hope our dialogue will help others with their studies. I would be happy to extend it if my point is still unclear.

- Andrew

TheNightKing · Answer

Hi Andrew, You made it absolutely clear. No further questions my lord, I rest my case

Keeping that aside, I was considering 15*15 as two parallel sides and was concluding figure 2 as resultant rectangle. But since 15*15 represents length*breadth then it is crystal clear what is happening here. Thank you for taking your time out and explaining this at length. Have a great day!

AndrewN · Answer

TheNightKing  - Yes, I suspected it might be a simple misinterpretation, but it is easy to do, and I am sure you are not alone in thinking of the problem in such a way. To turn the tables, I was once reading when I came across the word "underfed." I could not wrap my head around its meaning in the context because I was seeing it as an adjective based on the nonsensical verb "to derf." By the time I tracked down where the word would appear in the dictionary, my error had dawned on me. Sometimes it is hard to see something so obvious when your mind perceives the information in a certain way. And, getting back to the GMAT™, this is a test that will really punish you for reasoning that is even slightly off-center. I am glad my explanation sufficed, in this case.

Thank you, and, as you said in a manner of speaking, cheers.

- Andrew

frankgraves · Answer

I solved this problem by checking the answer choices.
Since the netting is 40 meters, the parameter should also be 40 meters.
Parameter of rectangular : 2 x (width + length)
Note that for squares, width = length.

A. 2 x (12+8) = 40 m - Okay
B. 2 x (15+8) = 46 m - Not okay
C. 2 x (10+10) = 40 m - Okay
D. 2 x (15+15) = 60 m - Not okay
E. 2 x (15+5) = 40 m - Okay

Eliminate B and D.
Now check the area of the remaining options A, C, E.

A. 12 x 8 = 96 m2
C. 10 x 10 = 100 m2
E. 15 x 5 = 75 m2

C has the largest area and hence is the correct answer.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

BottomJee · Answer

I think this is a  high-quality  question and I agree with explanation.

Ujjwalk31 · Answer

I did not quite understand the solution. To build a rectangular chicken pen, we are making square.

Bunuel · Answer

____________________________
Please check the discussion above.

M05-18

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