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

Hint:

A largest circle in a square of side s has area pi s squared over 4, whatever the size of the square. Add up the circle areas over all the small squares.

Answer 1

The Correct Option is C

Step 1: Scrap in one square:
In a square of side \( s \), the largest circle has diameter \( s \), so its area is \( \dfrac{\pi s^2}{4} \). The scrap is \( s^2 - \dfrac{\pi s^2}{4} = s^2\left(1-\dfrac{\pi}{4}\right) \), which is the fraction \( 1 - \dfrac{\pi}{4} \) of the square.

Step 2: First process:
The \( n \) small squares have sides \( s_1, \ldots, s_n \) with \( \sum s_i^2 = a^2 \) (they fill the original square). Total scrap \( = \sum s_i^2\left(1-\dfrac{\pi}{4}\right) = a^2\left(1-\dfrac{\pi}{4}\right) \).

Step 3: Second process:
One circle of diameter \( a \): scrap \( = a^2\left(1-\dfrac{\pi}{4}\right) \).

Step 4: Ratio:
The two scrap areas are equal, so the ratio is 1 : 1, whatever the number of pieces or their sizes.

Final Answer:
The correct option is (3): 1 : 1