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Question:

Consider two different cloth-cutting processes. In the first one, n circular cloth pieces are cut from a square cloth piece of side a in the following steps: the original square of side a is divided into n smaller squares, not necessarily of the same size; then a circle of maximum possible area is cut from each of the smaller squares. In the second process, only one circle of maximum possible area is cut from the square of side a and the process ends there. The cloth pieces remaining after cutting the circles are scrapped in both the processes. The ratio of the total area of scrap cloth generated in the former to that in the latter is:

Explanation:

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

A₁, A₂, A₃, A₄, A₅, ... , A_n

Also, let A₁ + A₂ + A₃ + A₄ + A₅ + ... + A_n = 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 $\frac{\pi A}{4}$

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

The area of the scrap material will be:

Area of Square - Area of the single maximum area Circle = A - $\left(\frac{\pi A}{4}\right)=\frac{A}{4}(4-\pi )$

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

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

$=\frac{\pi }{4}{A}_1+\frac{\pi }{4}{A}_2+\frac{\pi }{4}{A}_3+\cdots +\frac{\pi }{4}{A}_n=\frac{\pi }{4}(A_1+A_2+A_3+\cdots +A_n)=\frac{\pi }{4}A$

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

∴ Area of the scrap material will be:

$A-\frac{\pi }{4}A=\frac{A}{4}(4-\pi )$

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).