The ratio of the base radii of two right circular cones of the same height is 3 : 4. Find the ratio of their volumes.

Let radius of cones be 3a and 4a.

Since, height of both cones is same let it be h.

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

Ratio of volume of two cones = $\dfrac{\text{Volume of cone 1}}{\text{Volume of cone 2}}$

$\dfrac{\text{Volume of cone 1}}{\text{Volume of cone 2}} = \dfrac{\dfrac{1}{3}π \times (3a)^2 \times h}{\dfrac{1}{3}π \times (4a)^2 \times h} = \dfrac{(3a)^2}{(4a)^2} = \dfrac{9a^2}{16a^2} = \dfrac{9}{16}.$

Hence, the ratio of the volume of two cones = 9 : 16.