Discussion
A solid rubber sphere weighs 6kg when its diameter is 6cm. Using the same material, a hollow sphere is made with outer diameter 18cm and inner diameter 12 cm. What is its weight?
Answer: A
Same material means the same density, so weight is proportional to volume.
\[ V_{\text{solid}} = \frac{4}{3}\pi (3)^{3} = 36\pi \text{ cm}^{3} \;\longrightarrow\; 6 \text{ kg} \]
The hollow sphere has outer radius 9 cm and inner radius 6 cm:
\[ V_{\text{hollow}} = \frac{4}{3}\pi\left(9^{3} - 6^{3}\right) = \frac{4}{3}\pi (729 - 216) = \frac{4}{3}\pi (513) = 684\pi \text{ cm}^{3} \]
\[ \text{weight} = 6 \times \frac{684\pi}{36\pi} = 6 \times 19 = 114 \text{ kg} \]
The cavity must be subtracted; treating the object as a solid 18 cm sphere would give 162 kg.
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