A solid metallic cylinder of base radius $3\text{ cm}$ and height $5\text{ cm}$ is melted to form cones, each of height $1\text{ cm}$ and base radius $1\text{ mm}$. The number of cones is

### Concept & Unit Conversion in Volume When a large object is melted to form smaller identical objects, the number of smaller objects is the total original volume divided by the volume of one smaller object. **Crucially, all dimensions must be in the same units.** $$Number\ of\ Objects = \frac{Volume\ of\ Large\ Cylinder}{Volume\ of\ One\ Small\ Cone}$$ ### Step-by-Step Solution * **Step 1:** Convert all units to centimeters. Cylinder: $R = 3\text{ cm}$, $H = 5\text{ cm}$ Cone: height $h = 1\text{ cm}$, radius $r = 1\text{ mm} = 0.1\text{ cm} = \frac{1}{10}\text{ cm}$ * **Step 2:** Calculate the volume of the original cylinder. $V_{cylinder} = \pi R^2 H = \pi (3)^2 (5) = 45\pi\text{ cm}^3$ * **Step 3:** Calculate the volume of a single small cone. $V_{cone} = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi \left(\frac{1}{10}\right)^2 (1) = \frac{1}{3}\pi \left(\frac{1}{100}\right) = \frac{\pi}{300}\text{ cm}^3$ * **Step 4:** Divide the total volume by the single cone volume. $Number\ of\ cones = \frac{45\pi}{\frac{\pi}{300}}$ $Number\ of\ cones = 45 \times 300 = 13500$ ### Exam Strategy & Shortcut Write the volumes as a single combined fraction before simplifying: $\frac{\pi \times 3 \times 3 \times 5}{\frac{1}{3} \times \pi \times 0.1 \times 0.1 \times 1}$. Cancel the $\pi$ immediately. Numerator = $45$. Denominator = $\frac{0.01}{3} = \frac{1}{300}$. $45 \times 300 = 13500$. ### Common Pitfall Failing to convert the $1\text{ mm}$ radius to centimeters will yield $r = 1\text{ cm}$, leading to $V_{cone} = \frac{\pi}{3}$, resulting in an answer of $135$ (which isn't an option, but would cause panic). Always harmonize units first. ### Final Answer Therefore, the correct answer is **13500**.