What is the greatest number of rectangles with integer side lengths and perimeter 10 that can be cut from a piece of paper with width 24 and length 60?
to have perimeter 10, there are only two choices:
1x4
2x3
So, which of those will produce the greater number of rectangles?
To find the greatest number of rectangles with integer side lengths and perimeter 10 that can be cut from a piece of paper with width 24 and length 60, we will first determine the possible dimensions of the rectangles and then calculate how many rectangles can be formed.
1. Let's derive the possible dimensions of the rectangles by considering their perimeter and integer side lengths.
- The formula for the perimeter of a rectangle is: P = 2(length + width)
- In this case, we have P = 10. So, 10 = 2(length + width).
- We need to find pairs of integers (length, width) such that length + width = 5.
- The possible pairs are: (1, 4), (2, 3), (3, 2), (4, 1).
2. Now, we will calculate how many rectangles can be formed using these pairs of dimensions (length, width).
- For the pair (1, 4):
- We can use the width of 24 to form 6 rectangles with a length of 1.
- For the pair (2, 3):
- We can use the width of 24 to form 8 rectangles with a length of 2.
- For the pair (3, 2):
- We can use the width of 24 to form 12 rectangles with a length of 3.
- For the pair (4, 1):
- We can use the width of 24 to form 4 rectangles with a length of 4.
3. To find the total number of rectangles, we sum up the number of rectangles formed for each pair of dimensions:
- Total number of rectangles = 6 + 8 + 12 + 4 = 30.
Therefore, the greatest number of rectangles with integer side lengths and perimeter 10 that can be cut from a piece of paper with width 24 and length 60 is 30.