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.
Considering Pythagoras to the right angle
Same post answered by Steve
http://www.jiskha.com/display.cgi?id=1467634318
Apply Pythagoras to the right angle with
sides 2r, 12-2r and 9-2r.
This leads to the equation
4r2 – 85r + 225 =0
which solves to give r = 3.15
In the case where breadth is ‘b’ and the length is 2b, form the right angled triangle with sides
2r, 2b-2r and b-2r and proceed using the quadratic formula to get the solutions
r=5b/2 , b/2
The first solution is rejected.
how do you explain this? thanks in advanced
Hmmm. I originally rejected the idea that the centers would lie on the diagonal, because I thought the circles would overlap or not fill the maximum area.
I'm not sure how your equation comes about, but it appears you have solved the problem.
To find the maximum possible radius of the circles that can be cut from the given sheet of paper, we need to analyze the dimensions of the paper and determine the relationship between them.
Let's start by considering the given dimensions: a 12cm by 9cm sheet of paper. We want to cut out identical circles from this rectangular sheet.
To maximize the radius of the circles, we need to fit them into the sheet of paper as efficiently as possible. Since the circles are identical, they should touch each other without overlapping. This means we need to fit two circles side by side along the longer side of the paper.
Given the length of the paper is twice the breadth (9cm), we know that the length is 2 * 9cm = 18cm. We can visualize the paper as a rectangle with sides measuring 18cm by 9cm.
If we fit two identical circles side by side along the length (18cm), we can imagine that the combined width of the two circles should be equal to the breadth of the paper (9cm). To find the maximum possible radius, we divide the breadth by 2:
Radius = 9cm / 2 = 4.5cm
Hence, the maximum possible radius for the circles that can be cut from the given paper is 4.5cm.
Note: The explanation above assumes we can utilize the entire sheet of paper without any wastage. In practical situations, there may be some wastage due to the circular shape, cutting techniques, or other factors.