Think of various rectangles with perimeter 40 units (the sides do not have to be integers). Class 9

Think of various rectangles with perimeter 40 units (the sides do not have to be integers). Class 9

Question 1.

Think of various rectangles with perimeter 40 units (the sides do not have to be integers). Class 9

i. How many such rectangles are there?

ii. Among them, is there one whose area is the largest? What are its dimensions?

iii. 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?.

Answer:

Given: Perimeter of Rectangle = 40 units

2(l + b) = 40 units.

l + b = 20 units.

i. There are infinitely many such rectangles. Since the sides do not have to be integers, we can have dimensions such as 1 × 19, 2 × 18, 5 × 15, 9.5 × 10.5 and so on. If the two sides add up to 20, we get a valid rectangle.

ii. Considering different dimensions, we observe that when both sides are equal that is, when the rectangle becomes a square of side 10 units the area is 100 sQuestion units, which is the largest possible. When one side increases and the other decreases by the same amount, the area always becomes smaller.

iii. As one side becomes very small and the other becomes very large for example 0.1 × 19.9 or 0.01 × 19.99, the area keeps decreasing. It never actually reaches zero, but it can get as close to zero as we wish. Therefore, there is no rectangle with the smallest area.

Interestingly, among all rectangles with the same perimeter, it is the square that has the largest area. The most balanced shape always encloses the most space.