The radii of two cones are in the ratio 2:3 and their volumes are in the ratio 1:3. Then the ratio of their heights is:

Let the radii of the two cones be 2r and 3r, and their heights be h1 and h2.

The ratio of volumes of the two cones is 1:3

$$\dfrac{\frac{1}{3}\pi\left(2r\right)^2h_1}{\frac{1}{3}\pi\left(3r\right)^2h_2}=\dfrac{1}{3}$$

$$\dfrac{4r^2h_1}{9r^2h_2}=\dfrac{1}{3}$$

$$\dfrac{h_1}{h_2}=\dfrac{3}{4}$$

$$h_1:h_2=3:4$$

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