If the radius of the base and height of a cylinder and cone are each equal to $r$, and the radius of a hemisphere is also equal to $r$, then the volumes of the cone, cylinder and hemisphere are in the ratio

### Concept & Volume Comparison By standardizing all dimensions to a single variable $r$, we can directly compare the volumetric formulas of these three basic 3D shapes. Volume of a cone: $$V = \frac{1}{3}\pi r^2 h$$ Volume of a cylinder: $$V = \pi r^2 h$$ Volume of a hemisphere: $$V = \frac{2}{3}\pi r^3$$ ### Step-by-Step Solution 1. Express the volume of the cone, substituting $h = r$: $V_{cone} = \frac{1}{3}\pi r^2 (r) = \frac{1}{3}\pi r^3$. 2. Express the volume of the cylinder, substituting $h = r$: $V_{cyl} = \pi r^2 (r) = \pi r^3$. 3. Express the volume of the hemisphere, which already uses only $r$: $V_{hemi} = \frac{2}{3}\pi r^3$. 4. Form the ratio $V_{cone} : V_{cyl} : V_{hemi}$: $\frac{1}{3}\pi r^3 : \pi r^3 : \frac{2}{3}\pi r^3$. 5. Divide all terms by the common factor $\pi r^3$: $\frac{1}{3} : 1 : \frac{2}{3}$. 6. Multiply the entire ratio by 3 to remove fractions: $1 : 3 : 2$. ### Exam Strategy & Shortcut This is a fundamental geometric relationship discovered by Archimedes. A cone, hemisphere, and cylinder with the same radius and height always have a volume ratio of 1 : 2 : 3 respectively. Be careful to check the order asked in the question (cone, cylinder, hemisphere), which reorders it to 1 : 3 : 2. ### Common Pitfall Misreading the order requested in the prompt. The problem asks for "cone, cylinder, hemisphere", not the traditional "cone, hemisphere, cylinder" sequence which yields 1:2:3. ### Final Answer Therefore, the correct answer is **1 : 3 : 2**.