Two identical circles are to be cut from
a 12cm by 9cm sheet of paper.
What is the maximum possible radius of these circles?
Show that if the length of the sheet of paper is twice the breadth of the paper, then the radius of the largest circles which can be cut out is half of the breadth.
Sounds like the two circles' centers are on a line parallel with a side of the rectangle. In that case, the diameter of the circles is half the longest dimension of the rectangle.
So, two circles of radius 6 can be cut from the paper. Any bigger than that, and they won't both fit in the 12cm dimension.
To find the maximum possible radius of the circles that can be cut from a 12cm by 9cm sheet of paper, we need to consider the dimensions of the paper and the constraints of the problem.
First, let's visualize the situation. We have a rectangular sheet of paper with length 12cm and breadth 9cm. We want to cut out circles from this sheet, and the circles need to be identical.
To find the maximum possible radius, we need to make use of the fact that the circles need to fit inside the rectangle without overlapping. In other words, the diameter of each circle should be less than or equal to the shortest side of the rectangle.
Since we want to maximize the radius, we should choose the diameter to be equal to the length of one side of the rectangle. In this case, the diameter should be equal to the breadth of the paper (9cm).
The radius of a circle is half of its diameter, so the maximum possible radius for the circles that can be cut from the given sheet of paper is 9cm/2 = 4.5cm.
Now, let's prove the second statement: "If the length of the sheet of paper is twice the breadth of the paper, then the radius of the largest circles which can be cut out is half of the breadth."
If the length of the sheet is twice the breadth, then we have a rectangle with dimensions 2b by b, where b represents the breadth.
To find the maximum possible radius, we need to consider the relationship between the dimensions of the rectangle and the diameter of the circles.
Since the circles need to fit inside the rectangle without overlapping, the diameter of each circle should be less than or equal to the shortest side of the rectangle. In this case, the diameter should be equal to the breadth (b) of the rectangle.
The radius of a circle is half of its diameter, so the maximum possible radius for the circles that can be cut from the given rectangle is b/2.
Thus, we have proved that if the length of the sheet of paper is twice the breadth of the paper, then the radius of the largest circles which can be cut out is half of the breadth.