Think of various rectangles with perimeter 40 units (the sides do not have to be integers).
- How many such rectangles are there?
- Among them, is there one whose area is the largest? What are its dimensions?
- Among all these rectangles, is there one whose area is the smallest? What are its dimensions?
Do either of these answers come as a surprise to you?
There are infinitely many such rectangles. Since the sides do not have to be integers, any positive real number for length less than 20 will give a corresponding breadth , forming a valid rectangle with a perimeter of 40 units.
Yes, among these rectangles, there is one whose area is the largest. Its dimensions are 10 units by 10 units.
For a given perimeter, a square always encloses the maximum area. Here, the sum of length and breadth is 20, and the area
is maximized when , giving a maximum area of 100 sq. units.
No, among all these rectangles, there is no rectangle whose area is the smallest. As the length gets closer to 0, the area also gets closer to 0, but it can never actually reach 0 for a valid rectangle.
It might come as a surprise that a square gives the largest area, and that there is no minimum area possible.
Explanation
This problem explores the relationship between the perimeter and area of rectangles. Given a fixed perimeter of 40, the semi-perimeter is . Since and can be any positive real numbers, there are infinite combinations.
The area is a quadratic function
which has a maximum value at . Thus, the maximum area occurs when the rectangle is a square of side 10. The area approaches 0 as the dimensions become extremely thin, but never reaches 0, so there is no minimum area.