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.

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

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