### Instruction: Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles?### Response: I want to maximize the distance between the centers of the circles, so I will try to place them as far apart as possible within the rectangle.
One way to do this is to put one circle in a corner of the rectangle, and the other circle in the opposite corner.
The center of the circle in the corner will be 3 inches away from each edge of the rectangle, since the circle has a diameter of 6 inches.
The center of the other circle will be 3 inches away from the other two edges of the rectangle.
To find the distance between the centers, I can use the Pythagorean theorem, since the centers form a right triangle with the sides of the rectangle.
The hypotenuse of the right triangle is the distance I want to find.
The legs of the right triangle are the differences between the coordinates of the centers.
One leg is 14 - 6 = 8 inches long, since the centers are 6 inches apart along the longer side of the rectangle.
The other leg is 12 - 6 = 6 inches long, since the centers are 6 inches apart along the shorter side of the rectangle.