Multiple choice

Find the value of Q1 and Q2 and state the correct relationship.\nQuantity I: Area of rectangle whose sides are in the ratio 3 : 1 and perimeter twice the circumference of a semi-circle with area 77 cm2.\nQuantity II: Curved surface area of a cylinder of radius 12 cm and height equal to the side of square of area 49 cm2.

  1. Quantity I > Quantity II

  2. Quantity I ≥ Quantity II

  3. Quantity II > Quantity I

  4. Quantity II ≥ Quantity I

  5. Quantity I = Quantity II or Relation cannot be established
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Quantity I: Rectangle sides in ratio 3:1. Let sides be 3x and x. Perimeter = 2(3x + x) = 8x. Semi-circle with area 77 = (1/2)πr² gives r² = 49/π, r = 7/√π. Circumference = πr + 2r = π(7/√π) + 2(7/√π) = 7√π + 14/√π. Perimeter of rectangle = twice this = 14√π + 28/√π. So 8x = 14√π + 28/√π. Area = 3x² = 3[(14√π + 28/√π)/8]². This is complex. Let me recalculate: semi-circle area 77 means (πr²)/2 = 77, so r² = 154/π ≈ 49, so r ≈ 7. Semi-circle circumference (including diameter) = πr + 2r ≈ 7π + 14 ≈ 36. Rectangle perimeter = 2 × 36 = 72. So 8x = 72, x = 9. Rectangle area = 3x × x = 3 × 81 = 243. Quantity II: Square with area 49 has side 7. Cylinder height = 7, radius = 12. CSA = 2π × 12 × 7 = 168π ≈ 528. Since 528 > 243, Quantity II is greater.

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