A spherical ball of lead 6 cm in diameter is melted and recast into three spherical balls. The diameters of the balls are 2 cm, 4 cm and x cm. Which one of the following is correct?

  1. A. 5 \text{ cm} \lt x \text{ cm} \lt 5.2 \text{ cm}
  2. B. 5.2 \text{ cm} \lt x \text{ cm} \lt 5.4 \text{ cm} ✓
  3. C. 5.4 \text{ cm} \lt x \text{ cm} \lt 5.6 \text{ cm}
  4. D. 5.6 \text{ cm} \lt x \text{ cm} \lt 5.8 \text{ cm}

Correct Answer: B. 5.2 \text{ cm} \lt x \text{ cm} \lt 5.4 \text{ cm}

Explanation

Since volume is proportional to diameter cubed: 6^3 = 2^3 + 4^3 + x^3 \implies 216 = 8 + 64 + x^3 \implies x^3 = 144. We know 5^3 = 125 and 5.5^3 \approx 166.3. Checking values, 5.2^3 \approx 140.6 and 5.3^3 \approx 148.8. Thus, 5.2 \lt x \lt 5.4.

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