TheCATExam
Sign in

CAT 2003 Slot 2 — QA Question 41

Basics of Mensuration/PrismEasy
Passage / Data

Answer the following question based on the information given below.

A string of three English letters is formed as per the following rules:

  1. The first letter is any vowel.
  2. The second letter is m, n or p.
  3. If the second letter is m, then the third letter is any vowel which is different from the first letter.
  4. If the second letter is n, then the third letter is e or u.
    1. If the second letter is p, then the third letter is the same as the first letter.

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:

Answer & solution

  • 1 : 1

  • B

    2:1

  • C

    n(4-π)4n-π

  • D

    4n-πn(4-π)

Solution

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).

CAT 2003 Slot 2 QA Q41: Consider two different cloth-cutting processes. In the first one, n circular cloth pieces are cut from a squar — Solution | TheCATExam