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:

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

Hint:

In geometric optimization problems like this, calculate the total area of circles first, then subtract from the original area to find the scrap area.

Answer 1

The Correct Option is C

In the first process, the area of each smaller square is $\frac{a^2}{n}$, and the area of the circle that is cut from each smaller square is: \[ \text{Area of each circle} = \pi \left( \frac{a}{2\sqrt{n}} \right)^2 = \frac{\pi a^2}{4n} \] Thus, the total area of circles cut from all the squares is: \[ \text{Total area of circles} = n \times \frac{\pi a^2}{4n} = \frac{\pi a^2}{4} \] The total area of scrap cloth in the first process is: \[ \text{Area of scrap} = a^2 - \frac{\pi a^2}{4} = a^2 \left( 1 - \frac{\pi}{4} \right) \] In the second process, the area of the single circle cut from the square is: \[ \text{Area of circle} = \pi \left( \frac{a}{2} \right)^2 = \frac{\pi a^2}{4} \] The total area of scrap cloth in the second process is: \[ \text{Area of scrap} = a^2 - \frac{\pi a^2}{4} = a^2 \left( 1 - \frac{\pi}{4} \right) \] Thus, the ratio of the total scrap cloth generated in the first process to the second process is: \[ \text{Ratio} = \frac{n(4 - \pi)}{4n - \pi} \]