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 later is

• A. $1:1$
• B. $\sqrt{2}:1$
• C. $\frac{n(4-\pi )}{4n-\pi }$
• D. $\frac{4n-\pi }{n(4-\pi )}$

Answer 1

Answer: (A)

$1:1$

Consider the square of side x, then its area will be $x^2$.

The area of largest circle which can be cut from square is $=\pi {\left(\frac{x}{2}\right)}^2$$=\frac{\pi x^2}{4}$ .

Therefore, area scrapped $=x^2-\frac{\pi x^2}{4}=x^2(1-\frac{\pi }{4})$.

Hence,

The area scrapped / Area of square

$=\frac{x^2(1-\frac{\pi }{4})}{x^2}$

$=1-\frac{\pi }{4}=constant$.

As this ratio is constant, whether we cut a circle from smaller square or larger square, scrapped area will be a fixed percentage of square. Therefore, in this case where two squares are of the same size, the ratio will be 1 : 1.