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A hollow sphere made of material of density $$3\ \text{g/cm}^3$$ is floating with three fourth of its volume submerged in a liquid of density $$0.8\ \text{g/cm}^3$$. If the external volume of the sphere is $$200\ \text{cm}^3$$, find the volume of the cavity inside the sphere.
Correct Answer: 160
The sphere floats, so its weight equals the weight of liquid displaced, $$\frac{3}{4} \times 200 \times 0.8 = 120\ \text{g}$$. The material therefore has mass $$120\ \text{g}$$ and volume $$\frac{120}{3} = 40\ \text{cm}^3$$. The cavity volume is $$200 - 40 = 160\ \text{cm}^3$$.
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