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).
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Karishma
Owner of Angles and Arguments at https://anglesandarguments.com/
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