If the ratio of volumes of two cones is $2 : 3$ and the ratio of the radii of their bases is $1 : 2$, then the ratio of their heights will be

### Concept & Height Ratio We can isolate the height ratio in the volume formula of a cone by dividing the volume ratio by the square of the radius ratio. $$V = \frac{1}{3}\pi r^2 h \implies \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)$$ $$\frac{h_1}{h_2} = \frac{\frac{V_1}{V_2}}{\left(\frac{r_1}{r_2}\right)^2}$$ ### Step-by-Step Solution * **Step 1:** Extract the given ratios. Ratio of volumes ($V_1/V_2$) = $2/3$ Ratio of radii ($r_1/r_2$) = $1/2$ * **Step 2:** Substitute these values into the derived height ratio formula. $\frac{h_1}{h_2} = \frac{2/3}{(1/2)^2}$ * **Step 3:** Calculate the final ratio. $\frac{h_1}{h_2} = \frac{2/3}{1/4}$ $\frac{h_1}{h_2} = \frac{2}{3} \times \frac{4}{1} = \frac{8}{3}$ The ratio of their heights is $8 : 3$. ### Exam Strategy & Shortcut Write $V_1 : V_2$ as $2 : 3$. Write $r_1^2 : r_2^2$ as $1 : 4$. Since $h = V / r^2$ (ignoring constants), the height ratio is $(2/1) : (3/4)$. Multiply both sides by 4 to get $8 : 3$. ### Common Pitfall Dividing fractions incorrectly is the main trap here. Forgetting to flip the denominator fraction $(1/4)$ when multiplying will yield $1/6$, leading you completely astray. ### Final Answer Therefore, the correct answer is **8 : 3**.