Discussion

Explanation:

Let the areas of the n squares formed from the original square be:

A1, A2, A3, A4, A5, ... , An

Also, let A1 + A2 + A3 + A4 + A5 + ... + An = A                     ... (i)

Where, A will be the Area of the original square.

Now, if A = a2 (where a is a side of the square), then the area of the largest circle which can be drawn in it will have an area of π(a/2)2 = π/4 × a2 = π/4 × A

∴ Area of the maximum circle which can be cut from a square of area A is πA4

Case 1: When the cloth is cut using the 2nd process

The area of the scrap material will be:

Area of Square - Area of the single maximum area Circle = A - πA4=A4(4-π)

Case 2: When the cloth is cut using the 1st process

The sum of the areas of the maximum circles that can be cut out from the n squares

=π4A1+π4A2+π4A3++π4An=π4(A1+A2+A3++An)=π4A

Also, the sum of the areas of the n squares = Area of the original square = A

∴ Area of the scrap material will be:

A-π4A=A4(4-π)

From Cases 1 and 2, it is clear that the ratio of scrap left in the 1st process to the 2nd process is 1 : 1.

Hence, option (a).

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