The volumes of two cones are in the ratio 1:4 and their diameters are in the ratio 4: 5. What is the ratio of their heights?

Explanation

Volume of a cone V = \frac{1}{3}\pi r^2 h. The ratio of volumes \frac{V_1}{V_2} = (\frac{r_1}{r_2})^2 \times \frac{h_1}{h_2}. Given \frac{V_1}{V_2} = \frac{1}{4} and \frac{r_1}{r_2} = \frac{d_1}{d_2} = \frac{4}{5}. Thus, \frac{1}{4} = (\frac{4}{5})^2 \times \frac{h_1}{h_2} \Rightarrow \frac{1}{4} = \frac{16}{25} \times \frac{h_1}{h_2}. Solving for the height ratio gives \frac{h_1}{h_2} = \frac{25}{64}.

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