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Carpenter Rajesh Circular Plywood Cutouts

XAT 2012 · Quantitative Ability & Data Interpretation · Hard · Geometry

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Carpenter Rajesh has a circular piece of plywood of diameter 30 feet. He has cut out two disks of diameter 20 feet and 10 feet. What is the diameter of the largest disk that can be cut out from the remaining portion of the plywood piece?

  1. A.

    8.00 feet and ≤ 8.20 feet

  2. B.

    8.21 feet and ≤ 8.40 feet

  3. C.

    8.41 feet and ≤ 8.60 feet

  4. D.

    8.61 feet and ≤ 8.80 feet

  5. E.

    8.81 feet and ≤ 9.00 feet

Answer

C

Explanation

Radius of outer circle R=15R = 15. Radii of two cut circles: r1=10,r2=5r_1 = 10, r_2 = 5. Let radius of third maximum disk be xx. By applying Soddy's Circle Theorem or Descartes' Circle Theorem for internal and external tangents: x=607≈8.57x = \frac{60}{7} \approx 8.57 feet. Diameter = 2x=607≈8.572x = \frac{60}{7} \approx 8.57 feet, which lies between 8.41 and 8.60 feet.

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