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C.

(1) We know that the radius of the rug is 3ft (diameter is 6ft) and the radius of each of the stains is less than 1/10 ft. However we don't know how far the stains are from each other. INSUFFICIENT.

(2) We know that the centers of the stains are less than 4ft apart. Maximum distance between the 2 stains is less than 4 and 2/10 ft. [(2*1/10)+4]. However we do not know the size of the rug if it can completely cover 4 and 2/10 ft. INSUFFICIENT.

With (1) and (2), we know that the diameter of the rug (6ft) can completely cover both stains (max distance is 4 and 2/10 ft). SUFFICIENT.

PS: The information about the stains' location from the wall is unimportant since we do not know the dimensions of the room.

Originally posted by crza on 03 Jul 2017, 04:37.
Last edited by crza on 11 Jul 2017, 07:22, edited 1 time in total.
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AbdurRakib wrote:
Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than \(\frac{π}{100}\) square feet in area and each more than 3 feet from the nearest wall. Can the rug be placed to cover both stains ?

(1) Jack's rug covers an area of 9π square feet.

(2) The centers of the stains are less than 4 feet apart.


\(\frac{π}{100}\) square feet is the maximum area of each stain. So their radii are at most 1/10th of a foot.
Each is more than 3 feet from the wall so the distance between the circumference of the stain and the closest wall is at most 3 feet.
To find whether the rug covers the stains, we need two things - how big the rug is (think about the two extreme cases in which it is either very small or very big) and how far apart the stains are (to know how much area the rug must cover). Since we don't have to cover the whole room but just the stains, the actual size of the room doesn't matter to us (except if it gives us the maximum possible distance between the stains)


(1) Jack's rug covers an area of 9π square feet.
Now we know how big the rug is but not how far apart the stains are. The rug has a radius of 3 feet and hence a diameter of 6 feet.

(2) The centers of the stains are less than 4 feet apart.
We know how far apart the stains are but not how big the rug is.

Using both, we know how big the rug is (so the area it can cover) and we know what area will cover the two stains completely.
Centers of the stains are at most 4 feet apart. So to cover the stains completely, the rug should have a diameter of 4 + 1/10 + 1/10 = 4.2 feet or more. It does.
Sufficient.

Answer (C)

Note that had the rug been of a diameter less than 4.2 (say it were of diameter 3), could we have said that it WILL NOT cover the stains? No. Because we are given that 4 feet is the maximum distance between the two centers. The two centers could have very well been just 1 foot apart too. So in that case, answer would have been (E).

Check out the discussion on circles in this video: https://youtu.be/V_-N5Yi1UFo

Originally posted by KarishmaB on 18 Apr 2022, 22:38.
Last edited by KarishmaB on 06 Sep 2023, 23:42, edited 1 time in total.
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crza wrote:
C.

(1) We know that the radius of the rug is 3ft (diameter is 6ft) and the radius of each of the stains is 1/10 ft. However we don't know how far the stains are from each other. INSUFFICIENT.

(2) We know that the centers of the stains are less than 4ft apart. Maximum distance between the 2 stains is less than 4 and 2/10 ft. [(2*1/10)+4]. However we do not know the size of the rug if it can completely cover 4 and 2/10 ft. INSUFFICIENT.

With (1) and (2), we know that the diameter of the rug (6ft) can completely cover both stains (max distance is 4 and 2/10 ft). SUFFICIENT.

PS: The information about the stains' location from the wall is unimportant since we do not know the dimensions of the room.



Hi crza,

I think you missed out a keyword - Distance between the 2 stains is less than 4ft but more than 2/10 ft.
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ashygoyal wrote:
crza wrote:
C.

(1) We know that the radius of the rug is 3ft (diameter is 6ft) and the radius of each of the stains is 1/10 ft. However we don't know how far the stains are from each other. INSUFFICIENT.

(2) We know that the centers of the stains are less than 4ft apart. Maximum distance between the 2 stains is less than 4 and 2/10 ft. [(2*1/10)+4]. However we do not know the size of the rug if it can completely cover 4 and 2/10 ft. INSUFFICIENT.

With (1) and (2), we know that the diameter of the rug (6ft) can completely cover both stains (max distance is 4 and 2/10 ft). SUFFICIENT.

PS: The information about the stains' location from the wall is unimportant since we do not know the dimensions of the room.



Hi crza,

I think you missed out a keyword - Distance between the 2 stains is less than 4ft but more than 2/10 ft.


Hi ashygoyal,
In my statement above, I computed the maximum possible distance between the 2 stains as follows:
distance between the 2 stains = less than 4 ft
radius of each stain = 1/10 ft
For the purposes of computation, let us suppose the centers of the stains are located 4 ft apart.
Thus, the distance between the 2 stains from 1 end to the other end is 4ft + 1/10 (radius of one stain) + 1/10 (radius of the other stain) = 4 and 2/10 ft.
However, since we are told that the centers are located less than 4 ft apart, the maximum distance must be less than 4 and 2/10 ft.

In this question, we only care about the maximum distance between the stains.
In case you're interested, minimum possible distance between the stains is 4/10. This can be computed by adding the diameter (2/10) of the 2 stains.

Hope my explanation is clear.
Let me know if you have further clarifications! :D
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To follow up on some people's comment that the >3 feet distance from the nearest wall is redundant information: No, it's not.

As others have pointed out above, we can conclude from the combined statements (1) and (2) that the rug is large enough to cover both stains. However, the question is if the rug can be placed to cover both stains and to answer this, we must know that it's farther away from the wall than its radius. That is exactly why the 3 feet minimum distance is included in the question.

If the minimum distance was anything less than 3 feet, then the answer would actually be E.
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Re: Jack wants to use a circular rug on his rectangular office floor to co [#permalink]
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Max Radius of each stain is 1/10
Max Diameter of each stain is 1/10 + 1/10 = 2/5
Largest distance between center of stains is 4ft

Max diameter needed to cover is 4+ 1/10 + 1/10 = 21/5 = 4.2
Diameter of rug = 6

6>4.2 there for yes
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AbdurRakib wrote:
Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than \(\frac{π}{100}\) square feet in area and each more than 3 feet from the nearest wall. Can the rug be placed to cover both stains ?

(1) Jack's rug covers an area of 9π square feet.

(2) The centers of the stains are less than 4 feet apart.



Let radius of each stain = r
\(=> π * r^2 < π/100\)
\(=> r < 0.1\)feet

Statement 1: Area of rug \(= 9π = π * R^2\), where R is the radius of the rug
=> R = 3 feet

However, we do not know how far apart the stains are. If the stains are 2 feet apart, the rug can cover them. However, if the stains are 8 feet apart, the rug cannot cover them - Insufficient

Statement 2: The centers of the stains are less than 4 feet apart.
However, we do not know the radius of the stains. Also, we do not know the size of the rug - Insufficient


Combining:
The stains have radius 0.1 ft; and they are less than 4 feet apart
Thus, the maximum distance between the extreme points of the 2 stains is less than 0.1 + 4 + 0.1 = 4.2 ft
Since the radius of the rug is 3 ft (diameter is 6 ft), the rug can cover the stains.

However, it can cover the stains ONLY IF it were GEOMETRICALLY possible to place the rug over the stains. Since the stains are more than 3 feet away from the walls, there is no problem placing the rug. This is explained in the image below:

Attachment:
11.JPG
11.JPG [ 61.23 KiB | Viewed 41260 times ]



Answer C
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AbdurRakib wrote:
Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than \(\frac{π}{100}\) square feet in area and each more than 3 feet from the nearest wall. Can the rug be placed to cover both stains ?

(1) Jack's rug covers an area of 9π square feet.

(2) The centers of the stains are less than 4 feet apart.


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Hi everyone,

Just a small doubt: 'Jack wants to use a circular rug on his rectangular office floor' - Here, the question just mentioned that Jack wants to use a circular rug on a rectangular office, whose size is unknown. What if the rectangular office area is so big that the area of the rug is negligible & the stains might lie somewhere else. Am I missing anything here?

Thank you!
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SJKC wrote:
Hi everyone,

Just a small doubt: 'Jack wants to use a circular rug on his rectangular office floor' - Here, the question just mentioned that Jack wants to use a circular rug on a rectangular office, whose size is unknown. What if the rectangular office area is so big that the area of the rug is negligible & the stains might lie somewhere else. Am I missing anything here?

Thank you!



Rectangular office can have any area, but we know the stains lie less than 4 feet apart from each other, and we have to just place the circular rug on top of these stains.

Thus the area of office is immaterial and we are concerned with the area of rug and the maximum distance between the stains to check whether rug can cover them.
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Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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Re: Jack wants to use a circular rug on his rectangular office floor to co [#permalink]
chetan2u wrote:
SJKC wrote:
Hi everyone,

Just a small doubt: 'Jack wants to use a circular rug on his rectangular office floor' - Here, the question just mentioned that Jack wants to use a circular rug on a rectangular office, whose size is unknown. What if the rectangular office area is so big that the area of the rug is negligible & the stains might lie somewhere else. Am I missing anything here?

Thank you!



Rectangular office can have any area, but we know the stains lie less than 4 feet apart from each other, and we have to just place the circular rug on top of these stains.

Thus the area of office is immaterial and we are concerned with the area of rug and the maximum distance between the stains to check whether rug can cover them.


I got the same doubt but I still didn't understand how the area of Rectangular office is not necessary here ? From your words
Quote:
"we have to just place the circular rug on top of these stains. "

what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

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lostminer wrote:
what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

lostminer we know the stains lie less than 4 feet apart from each other

Posted from my mobile device
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Re: Jack wants to use a circular rug on his rectangular office floor to co [#permalink]
avigutman wrote:
lostminer wrote:
what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

lostminer we know the stains lie less than 4 feet apart from each other

Posted from my mobile device


avigutman please explain why the fourth case is not possible. where am I going wrong ?
Attachments

Screenshot 2022-04-18 180308.png
Screenshot 2022-04-18 180308.png [ 161.68 KiB | Viewed 16214 times ]

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lostminer wrote:
avigutman wrote:
lostminer wrote:
what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

lostminer we know the stains lie less than 4 feet apart from each other

Posted from my mobile device


avigutman please explain why the fourth case is not possible. where am I going wrong ?

The fourth case is possible, but the question asks not whether Jack WILL cover the stains but whether the rug CAN be placed to cover the stains.
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MartyTargetTestPrep wrote:
The fourth case is possible, but the question asks not whether Jack WILL cover the stains but whether the rug CAN be placed to cover the stains.

Precisely. The circular rug could be moved to cover both stains, regardless of the dimensions of the room. The language of the question stem makes all the difference. You have to be meticulous about your approach to DS questions, lostminer, or you can down several minutes figuring out the answer to a question that is not being asked. (This was one of my issues when I started my own preparation for the exam.) It is good practice to check the question stem, write down what you need to solve, work through the problem, and then check the question stem again, just to be confident in your answer. (GMATNinja refers to this process as "good hygiene.")

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lostminer wrote:
avigutman wrote:
what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

we know the stains lie less than 4 feet apart from each other

I had a hard time understanding your question, lostminer, because of your use of the word unable (boldfaced in your quote).
Your use of that word implies to me that you did understand the importance of the word can in the question stem. By the way, that word goes hand in hand with the word wants in the question stem.
Looking at the image that you uploaded, in which the rug sits next to the stains rather than on top of them, can you see that unable doesn't fit that picture?
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