Jack wants to use a circular rug on his rectangular office floor to co

Question

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.

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

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

chetan2u · Accepted Answer

Hi.. look at the att fig.. circle1.png the distance of 3' from the wall does not really affect the answer except when the stains are in the corner in a way that the circular rug does not reach the ends Solution when statements combined bigger circle is the circular rug an the smaller circle are the stains... it is given that stains are pi/100 in square feet, so pi*r_1^2=\frac{pi}{100}...r_1=\frac{1}{10} statement II gives the distance between them so the radius of the circular rug required is \frac{21}{10} statement I tells us that the area of the circular rug as pi*9 so pi*r_2^2=9pi...r_2=3 .. so the radius required is 21/10 and available is 3... hence it will cover C

JeffTargetTestPrep · Answer

Statement One Alone:
 
Jack's rug covers an area of 9π square feet.

If the area of the rug is 9π square feet, then the radius of the rug is 3 feet, since 3^2 x π = 9π. However, since we don’t know the distance between the centers of the two stains, we can’t determine whether the rug can cover both stains. For example, if the distance between the centers of the two stains is 1 foot, then the rug is big enough to cover them. However, if the distance between the centers of the two stains is 10 feet, then the rug is not big enough to cover them. Statement one alone is not sufficient to answer the question.

Statement Two Alone:
 
The centers of the stains are less than 4 feet apart.

Without knowing the area of the rug, we can’t determine whether the rug can cover both stains. For example, if the area of the rug is π square feet, then the rug is not big enough to cover them. However, if the area of the rug is 100π square feet, then the rug is big enough to cover them. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statement one, we know that the radius of the rug is 3 feet, and hence the diameter of the rug is 6 feet. Using statement two, we know that the centers of the stains are less than 4 feet apart. Moreover, we know that each of the stains has an area of less than π/100 square feet, and therefore the radius of each of the stains is less than 1/10 foot. Thus, if we place the rug so that the stains lie on a diameter of the rug, the rug is big enough to cover the stains.

Answer: C

crza · Answer

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.  . 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.

KarishmaB · Answer

\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

ashygoyal · Answer

Hi crza,

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

crza · Answer

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

danpo · Answer

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.

fireagablast · Answer

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

sujoykrdatta · Answer

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: 11.JPG Answer C

GMATinsight · Answer

Answer: Option C

Video solution by  GMATinsight

https://www.youtube.com/watch?v=vI_PoVNU6Es

SJKC · Answer

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!

chetan2u · Answer

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.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=FREYE7U-6LY

lostminer · Answer

I got the same doubt but I still didn't understand how the area of Rectangular office is not necessary here ? From your words 
what if we are unable to place the rug on both the stains ? the answer can be E as well right? Please explain.

GMATNinja   AndrewN   KarishmaB   ScottTargetTestPrep   MartyTargetTestPrep   avigutman   Vinit800HBS

avigutman · Answer

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

Posted from my mobile device

lostminer · Answer

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

MartyTargetTestPrep · Answer

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.

AndrewN · Answer

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.")

- Andrew

avigutman · Answer

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?

Jack wants to use a circular rug on his rectangular office floor to co

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