In designing pistons, two cylinders have equal heights but one has twice the diameter of the other. What is the ratio of their volumes?
1:4
The volume of a right circular cylinder is \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) the height.
Both cylinders have equal heights, so \(h\) is a common constant and the volume varies only with \(r^2\).
One cylinder has twice the diameter of the other; since radius is half the diameter, the radii are also in the ratio \(r_1 : r_2 = 1 : 2\).
Therefore the volume ratio is \(\dfrac{V_1}{V_2} = \dfrac{\pi r_1^2 h}{\pi r_2^2 h} = \dfrac{r_1^2}{r_2^2}\).
Substituting: \(\dfrac{r_1^2}{r_2^2} = \dfrac{1^2}{2^2} = \dfrac{1}{4}\), so \(V_1 : V_2 = 1 : 4\).
Key concept: for equal heights, cylinder volume is proportional to the square of the radius (or diameter); doubling the diameter quadruples the volume.
Hence the ratio of their volumes is 1:4.
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