Discussion
The internal and external radii of a hollow metallic spherical shell are 3 cm and 5 cm, respectively. Twenty eight such shells are melted and recast into a solid sphere. What is the surface area (in \(cm^{2}\)) of the new sphere, so formed?
Answer: B
The surface area is 784 pi square centimetres. The material in one shell is \[\frac{4}{3}\pi(5^3 - 3^3)\] \[= \frac{4}{3}\pi \times 98\] For 28 shells it is \[\frac{4}{3}\pi \times 98 \times 28\] \[= \frac{4}{3}\pi \times 2744\] The new solid sphere therefore has \(r^3 = 2744\), so r = 14 cm. Its surface area is \[4\pi r^2 = 4\pi \times 196 = 784\pi\] The other options correspond to radii of 12, 13 and 15 cm, which do not give 2744 as r cubed (12 cubed is 1728, 13 cubed is 2197 and 15 cubed is 3375).
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