Discussion
A spherical ball of lead, 3 cm in diameter is melted and recast into three spherical ball. The diameter of two of these are 1.5 cm and 2 cm. what is the diameter of third ball:
Answer: A
The third ball has a diameter of 2.5 cm.
Volume is conserved when the ball is recast, and a sphere's volume is proportional to the cube of its diameter, so the cubes of the diameters add up:
\[ \begin{aligned} 3^{3} &= 1.5^{3} + 2^{3} + d^{3} \\ 27 &= 3.375 + 8 + d^{3} \\ d^{3} &= 15.625 \\ d &= 2.5\ \text{cm} \end{aligned} \]
Treating diameters as if they simply added would give \(3 - 1.5 - 2 \lt 0\), which is impossible, and squaring instead of cubing gives about 1.66 cm, not offered. 3.6 cm is larger than the original ball, which cannot be.
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