Discussion
Two metallic right circular cones having their heights 4.1 cm and 4.3 cm and the radii of their bases 2.1 cm each, have been melted together and recast into a sphere. Find the diameter of the sphere.
Answer: C
Melting conserves volume, and both cones share the same base radius, so their volumes add through the heights.
\[ V = \frac{1}{3}\pi (2.1)^{2}(4.1 + 4.3) = \frac{1}{3}\pi (4.41)(8.4) \]
\[ = \frac{1}{3}\pi (37.044) = 12.348\pi \text{ cm}^{3} \]
Setting this equal to the sphere's volume:
\[ \frac{4}{3}\pi R^{3} = 12.348\pi \quad\Longrightarrow\quad R^{3} = 9.261 \]
\[ R = 2.1 \text{ cm} \quad\Longrightarrow\quad d = 4.2 \text{ cm} \]
The question asks for the diameter, so the radius must be doubled.
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