A rectangle with integer side lengths has perimeter 10. What is the gr

Maximize the number of rectangles by minimizing the size of the rectangles (area)

P=10
2L + 2W = 10 --> min area when difference in length and width is greatest
L=4 W=1

Area rectangle = 4*1

60*24/4 = 60*6 = 360 rectangles

If x and y are positive integers such that their sum is a CONSTANT:
The GREATEST possible value of xy will be yielded when x and y are AS CLOSE TO EACH OTHER AS POSSIBLE.
The LEAST possible value of xy will be yielded when x and y are AS FAR FROM EACH OTHER AS POSSIBLE.
Given x+y=100, where x and y are positive integers:
Greatest option for xy = 50*50 = 2500
Least option for xy = 1*99 = 99
In short:
To MAXIMIZE the product, MINIMIZE the difference.
To MINIMIZE the product, MAXIMIZE the difference.



A rectangle with integer side lengths has perimeter 10:
2L + 2W = 10
L+W = 5

What is the greatest number of these rectangles that can be cut from a piece of paper with width 24 and length 60?
To maximize the number of rectangles, we must minimize the value of LW -- the area of each rectangle.
As noted above:
To minimize the product of L and W, we must maximize the difference between L and W, as follows:
L=4, W=1, area = 4*1 = 4
Thus:
Greatest number of rectangles \(= \frac{paper-area}{rectangle-area} = \frac{24*60}{4} = 360\)

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