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# M05-18

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Math Expert
Joined: 02 Sep 2009
Posts: 49271

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16 Sep 2014, 00:25
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Difficulty:

5% (low)

Question Stats:

79% (00:33) correct 21% (00:36) wrong based on 174 sessions

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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

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Joined: 02 Sep 2009
Posts: 49271

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16 Sep 2014, 00:25
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:

Say the length and the width of the rectangle are $$x$$ and $$y$$.

Given: $$\text{Perimeter}=2x+2y=40$$. Reduce by 2: $$x+y=20$$. We have to maximize the area, so maximize the value of $$xy$$.

Useful property: for given sum of two numbers, their product is maximized when they are equal. Hence, the value of $$xy$$ will be maximized for $$x=y=10$$.

Approach 2:

This question can be easily solved if one knows the following property: a square has a larger area than any other quadrilateral with the same perimeter.

So, in order to maximize the area our rectangle must be a square: $$\text{Perimeter}=4x=40$$. Therefore $$x=10$$.

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Joined: 05 Aug 2015
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13 Mar 2016, 00:12
Bunuel wrote:
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:

Say the length and the width of the rectangle are $$x$$ and $$y$$.

Given: $$\text{Perimeter}=2x+2y=40$$. Reduce by 2: $$x+y=20$$. We have to maximize the area, so maximize the value of $$xy$$.

Useful property: for given sum of two numbers, their product is maximized when they are equal. Hence, the value of $$xy$$ will be maximized for $$x=y=10$$.

Approach 2:

This question can be easily solved if one knows the following property: a square has a larger area than any other quadrilateral with the same perimeter.

So, in order to maximize the area our rectangle must be a square: $$\text{Perimeter}=4x=40$$. Therefore $$x=10$$.

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.
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Math Expert
Joined: 02 Sep 2009
Posts: 49271

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13 Mar 2016, 08:17
1
happyface101 wrote:
Bunuel wrote:
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:

Say the length and the width of the rectangle are $$x$$ and $$y$$.

Given: $$\text{Perimeter}=2x+2y=40$$. Reduce by 2: $$x+y=20$$. We have to maximize the area, so maximize the value of $$xy$$.

Useful property: for given sum of two numbers, their product is maximized when they are equal. Hence, the value of $$xy$$ will be maximized for $$x=y=10$$.

Approach 2:

This question can be easily solved if one knows the following property: a square has a larger area than any other quadrilateral with the same perimeter.

So, in order to maximize the area our rectangle must be a square: $$\text{Perimeter}=4x=40$$. Therefore $$x=10$$.

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.

Yes, every square is a rectangle but not vise-versa.
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Joined: 26 Dec 2016
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28 Feb 2017, 06:59
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
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Joined: 22 May 2016
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25 Apr 2017, 05:58
BoomHH wrote:
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

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.
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20 Jul 2018, 23:36
Every sq is quadrilateral, but every sq is also a rectangle. Very simple but a good one to confuse :S

Wiki -
In Euclidean plane geometry, a rectangle is a quadrilateral with four right angles. It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle.

Another definition:
A rectangle is a shape with four sides and four corners. The corners are all right angles. It follows that the lengths of the pairs of sides opposite each other must be equal. ... A rectangle with all four sides equal in length is called a square.
Re: M05-18 &nbs [#permalink] 20 Jul 2018, 23:36
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# M05-18

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