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
Answer CYour sketch shows the stains to be more than 3 feet from top and bottom walls, shouldn't they also be more than 3 feet apart from the left and right walls as well ?