Page 173 - Ganit Kaushal

(a) The rectangle (24 × 1) has the greatest perimeter = 50 units.

(b) The rectangle (4 × 6) has the least perimeter = 20 units.
(c)  32 = 1 × 32; Perimeter of Rectangle = 2(1 + 32) = 66 units
Similarly, 32 = 2 × 16; Perimeter of Rectangle = 36 units

and 32 = 4 × 8; Perimeter of Rectangle = 24 units
So, we conclude that the rectangle (32 × 1) has the greatest perimeter = 66 units
and the rectangle (8 × 4) has the least perimeter = 24 units.
Ex 3. 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 answer.

Sol. Yes! For a given area A, it is possible to predict the shape of the rectangle with the greatest perimeter
as well as the least perimeter. Follow the procedure below:

(a) Write all the factors of the given area A in ascending or descending order.
(b) The rectangle with extreme factor pairs as its dimensions will have the greatest perimeter, and the
rectangle with the middle factor pairs as its dimensions will have the least perimeter.
For example: Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24; Extreme Factors = 1, 24; Middle Factors = 4, 6

\ For a given area of 24 sq units, the rectangle (24 × 1) has the greatest perimeter, and the rectangle
(6 × 4) has the least perimeter. (see Ex. 2 above)

For example: Factors of 32 = 1, 2, 4, 8, 16, 32; Extreme Factors = 1, 32; Middle Factors = 4, 8
\ For a given area of 32 sq units, the rectangle (32 × 1) has the greatest perimeter and the rectangle
(8 × 4) has the least perimeter. (see Ex. 2 above)
Thus, we can conclude that:
‘For a given area A, a rectangle with dimensions (A × 1) has the greatest A-unit
perimeter, and the rectangle with dimensions formed by the middle 1-unit
factors of A (when factors are written in ascending or descending order)
has the least perimeter’.
1
Ex 4. Avneet buys 9 square paving slabs, each with a side of m. He lays them in the form of a square.
2
(a) What is the perimeter of his arrangement? [See fig. (i)]
(b) Vineet does not like this arrangement, so he gets him to lay
them out like a cross. What is the perimeter of this
arrangement? [See fig. (ii)]
(c) Which arrangement has a greater perimeter?
(d) Avneet wonders if there is a way of getting an even greater (i) (ii)
perimeter. Can you find a way of doing this? (The paving slabs must meet along complete edges,
i.e., they cannot be broken.)
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Sol. (a) The square paving slabs are laid in the form of a square. Side of each square slab = m.
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 1 1 1 3
Each side of the square formed =  + +  m= m .
 2 2 2 2
3
Perimeter of his arrangement i.e., perimeter of square = 4 × side = 4 × m = 6 m.
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Perimeter and Area 171