Question (Let's explore - Page 141, 142) - Let's explore - Page 141 - Chapter 6 Class 6 - Perimeter and Area (Ganita Prakash)

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Question (Let's explore - Page 141, 142) On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. (a) Which rectangle has the greatest perimeter? (b) Which rectangle has the least perimeter? (c) If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answerAlright, let’s answer them one by one For an Area of 24 Square Units: The possible rectangles with whole number sides are: 1 unit × 24 units 2 units × 12 units 3 units × 8 units 4 units × 6 units a. Greatest Perimeter: The 1 × 24 rectangle has the greatest perimeter. Perimeter = 2 × (1 + 24) = 50 units. b. Least Perimeter: The 4 × 6 rectangle has the least perimeter. Perimeter = 2 × (4 + 6) = 20 units. For an Area of 32 sq cm: The possible rectangles with whole number sides are: 1 cm × 32 cm 2 cm × 16 cm 4 cm × 8 cm c. The answers would be: Greatest Perimeter: The 1 × 32 rectangle (Perimeter = 66 cm). Least Perimeter: The 4 × 8 rectangle (Perimeter = 24 cm). Predicting the Shape for Greatest and Least Perimeter: Yes, for any given area, it is possible to predict the shape of the rectangle with the greatest and least perimeter. Greatest Perimeter: The rectangle that is the longest and thinnest will always have the greatest perimeter. This corresponds to the dimensions of 1 and the area itself (e.g., 1 × 24 or 1 × 32). Least Perimeter: The rectangle that is the most "square-like" will always have the least perimeter. This corresponds to the dimensions with factors that are closest to each other (e.g., 4 × 6 for area 24; 4 × 8 for area 32).