A solid metallic sphere of radius R is heated so that its radius increases by 10%. Find the approximate percentage increase in its volume.
33.1%
The volume of a sphere is \(V = \dfrac{4}{3}\pi R^3\), so the volume is proportional to the cube of the radius: \(V \propto R^3\).
The radius increases by 10%, so the new radius is \(R' = R + 0.10R = 1.1R\).
Because volume goes as the cube, the volume scale factor is \(\left(\dfrac{R'}{R}\right)^3 = (1.1)^3\).
Evaluate: \((1.1)^3 = 1.1 \times 1.1 \times 1.1 = 1.331\), so the new volume is 1.331 times the original.
The fractional increase is \(1.331 - 1 = 0.331\) of the original volume.
As a percentage: \(0.331 \times 100 = 33.1\%\).
Key concept: when a linear dimension changes by a factor, volume changes by the cube of that factor. Hence the volume increases by about 33.1%.
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