The ratio of the heights of two right circular cones is 5 : 2 and that of their base radii is 2 : 5. Find the ratio of their volumes.

Let height of cones be 5a and 2a and radius of cones be 2b and 5b.

We know volume of cone = $\dfrac{1}{3}πr^2h$.

Ratio of volume of two cones = $\dfrac{\text{Vol. of Cone 1}}{\text{Vol. of Cone 2}}$

$\dfrac{\text{Vol. of Cone 1}}{\text{Vol. of Cone 2}} = \dfrac{\dfrac{1}{3}π \times (2b)^2 \times 5a}{\dfrac{1}{3}π \times (5b)^2 \times 2a} \\[1em] = \dfrac{4b^2 \times 5a}{25b^2 \times 2a} \\[1em] = \dfrac{20ab^2}{50ab^2} \\[1em] = \dfrac{2}{5}.$

Hence, the ratio of the volume of two cones = 2 : 5.