Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than 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 9Ï€ square feet.
(2) The centers of the stains are less than 4 feet apart.
Official Guide question
Answer: C
Jack wants to use a circular rug on his rectangular office
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The situation is depicted in the image below.jjjinapinch wrote:Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than 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 9Ï€ square feet.
(2) The centers of the stains are less than 4 feet apart.
Official Guide question
Answer: C
Since the area of each stain is less than π/100 sq feet, we have
A < π/100 sq feet; A = Area of each stain
πr^2 < π/100; r = Radius of each stain
=> r < 1/10 or d < 1/5; d = diametere of each stain
Statement 1: Rug covers an area of 9Ï€ sq feet
Area of rug = 9π sq feet = πR^2; R = Radius of the rug
=> R = 3 feet or D = 6 feet; D = diametere of the rug
Since we do not know how far the stains are from each other, we cannot determine whether the rug can cover both the stains. If the stains are too close to each other, the rug can cover both the stains since (twice the diameter of the stains = 2*1/5 = 2/5) < (The diameter of the rug = 6). Insufficient.
Statement 2: The centers of the stains are less than 4 feet apart.
Let's take the extreme case. Let's assume that the centers of the stains 4 feet apart and their area = π/100 sq feet each.
Thus, the maximum distance between the two stains = 4 + the diameters of the stains = 4 + 1/5 = 4.20 feet
We do not know the diameter of the rug. Insufficient.
Statement 1 & 2:
Since 4.20 feet (Maximum distance between the stains to be covered) < 6 feet (Diameter of the rug), the rug can cover both the stains. Sufficient.
The correct answer: C
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Statement One Alone:jjjinapinch wrote:Jack wants to use a circular rug on his rectangular office floor to cover two small circular stains, each less than 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 9Ï€ square feet.
(2) The centers of the stains are less than 4 feet apart.
Jack's rug covers an area of 9Ï€ 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
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