Problem:
Ms. Osborne asks each student in her class to draw a rectangle with integer side lengths and a perimeter of units. All of her students calculate the area of the rectangle they draw. What is the difference between the largest and smallest possible areas of the rectangles?
Answer Choices:
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Solution:
The formula for the perimeter of a rectangle is , so , and . Make a chart of the possible widths, lengths, and areas, assuming all the widths are shorter than all the lengths.
The largest possible area is and the smallest is , for a difference of square units.
Note: The product of two numbers with a fixed sum increases as the numbers get closer together. That means, given the same perimeter, the square has a larger area than any rectangle, and a rectangle with a shape closest to a square will have a larger area than other rectangles with equal perimeters.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions