A right circular cone is divided into two portions by a plane parallel to the base and passing through a point which is $\frac{1}{3}$rd of the height from the top. The ratio of the volume of the smaller cone to that of the remaining frustum of the cone is

### Concept & Volume Ratio of Similar Cones When a cone is cut by a plane parallel to its base, the smaller cone formed at the top is geometrically similar to the original large cone. The ratio of their volumes is the cube of the ratio of their corresponding linear dimensions (such as height or radius). $$ \frac{V_{\text{small}}}{V_{\text{large}}} = \left(\frac{h_{\text{small}}}{h_{\text{large}}}\right)^3 $$ ### Step-by-Step Solution - Let the height of the original large cone be $h$. - The cut is made at $\frac{1}{3}$rd of the height from the top, so the height of the smaller cone is $h_{\text{small}} = \frac{h}{3}$. - The ratio of the height of the smaller cone to the larger cone is $\frac{1}{3}$. - Therefore, the ratio of their volumes is $\left(\frac{1}{3}\right)^3 = \frac{1}{27}$. - Let the volume of the smaller cone be $1V$. The total volume of the large cone is $27V$. - The remaining part of the cone is the frustum. Its volume is the total volume minus the volume of the smaller cone: $V_{\text{frustum}} = 27V - 1V = 26V$. - The ratio of the volume of the smaller cone to the frustum is $\frac{1V}{26V} = 1 : 26$. ### Exam Strategy & Shortcut If a cone is cut parallel to its base at a ratio of $1/n$ of its height, the top cone's volume is $1/n^3$ of the total volume, and the frustum is the rest ($n^3 - 1$). Here, $n=3$, so the ratio is $1 : (3^3 - 1) = 1 : 26$. ### Common Pitfall Calculating the ratio of the small cone to the *entire* large cone ($1:27$) instead of the small cone to the *remaining frustum* ($1:26$). Always read carefully to see which two parts are being compared. ### Final Answer Therefore, the correct answer is **1 : 26**.