The radii of two cones are in the ratio $2 : 1$, their volumes are equal. Find the ratio of their heights.

### Concept & Volume Proportion When the volumes of two cones are equal, their heights are inversely proportional to the square of their radii. $$V = \frac{1}{3}\pi r^2 h$$ $$V_1 = V_2 \implies r_1^2 h_1 = r_2^2 h_2$$ ### Step-by-Step Solution * **Step 1:** Extract the given information. Ratio of radii ($r_1/r_2$) = $2/1$ Volumes are equal: $V_1 = V_2$ * **Step 2:** Setup the equation using the volume formula. $\frac{1}{3}\pi r_1^2 h_1 = \frac{1}{3}\pi r_2^2 h_2$ * **Step 3:** Simplify to find the height ratio. $r_1^2 h_1 = r_2^2 h_2$ $\frac{h_1}{h_2} = \frac{r_2^2}{r_1^2} = \left(\frac{r_2}{r_1}\right)^2$ * **Step 4:** Substitute the given radii ratio. Since $r_1/r_2 = 2/1$, then $r_2/r_1 = 1/2$. $\frac{h_1}{h_2} = \left(\frac{1}{2}\right)^2 = \frac{1}{4}$ The ratio of their heights is $1 : 4$. ### Exam Strategy & Shortcut If Volume is constant, $h \propto \frac{1}{r^2}$. Given $r$ ratio is $2 : 1$, square it to get $4 : 1$. Since $h$ is inversely proportional to $r^2$, invert the squared ratio to get $1 : 4$. ### Common Pitfall A common mistake is forgetting to square the radius ratio and simply stating that the height ratio is inversely proportional to the radius ($1 : 2$), which leads to an incorrect answer. ### Final Answer Therefore, the correct answer is **1 : 4**.