A right cylindrical vessel is full of water. How many right cones having the same radius and height as those of the right cylinder will be needed to store that water?

### Concept & Volume Ratio The volume of a cylinder is strictly three times the volume of a cone provided they share the exact same base radius and height. $$V_{cylinder} = \pi r^2 h$$ $$V_{cone} = \frac{1}{3}\pi r^2 h$$ ### Step-by-Step Solution * **Step 1:** Write the expression for the total volume of water to be stored (Volume of the cylinder). $Total\ Water = \pi r^2 h$ * **Step 2:** Write the expression for the capacity of a single cone. $Cone\ Capacity = \frac{1}{3}\pi r^2 h$ * **Step 3:** Divide the total water volume by the capacity of one cone to find the number of cones needed. $Number\ of\ cones = \frac{\pi r^2 h}{\frac{1}{3}\pi r^2 h}$ * **Step 4:** Simplify the fraction. The $\pi r^2 h$ terms cancel out completely. $Number\ of\ cones = \frac{1}{\frac{1}{3}} = 3$ ### Exam Strategy & Shortcut This is a fundamental geometric property. A cylinder can be perfectly divided into three equal-volume cones of the same base and height. No calculation is needed if you have this property memorized. ### Common Pitfall Some might confuse surface area relationships with volume relationships, or mistakenly think a cylinder holds twice the volume of a cone instead of three times. ### Final Answer Therefore, the correct answer is **3**.