A metallic sphere of radius 10.5 cm is melted and recast into small right circular cones, each of base radius 3.5 cm and height 3 cm. The number of cones so formed is

### Concept & Recasting Objects When a single large object is recast into multiple identical smaller objects, the total volume is conserved. The number of smaller objects is the total initial volume divided by the volume of one smaller object. ### Step-by-Step Solution 1. **Given:** Sphere radius ($R$) = $10.5\text{ cm}$. Cone radius ($r$) = $3.5\text{ cm}$, height ($h$) = $3\text{ cm}$. 2. **Set up the Equation:** Let $n$ be the number of cones. $V_{sphere} = n \times V_{cone}$ $\frac{4}{3}\pi R^3 = n \times (\frac{1}{3}\pi r^2h)$ 3. **Substitute and Simplify:** Cancel out $\frac{\pi}{3}$ from both sides: $4 \times (10.5)^3 = n \times (3.5)^2 \times 3$ $n = \frac{4 \times 10.5 \times 10.5 \times 10.5}{3.5 \times 3.5 \times 3}$ 4. **Calculate:** Notice that $10.5 / 3.5 = 3$. $n = \frac{4 \times (3 \times 3.5) \times (3 \times 3.5) \times 10.5}{3.5 \times 3.5 \times 3}$ $n = \frac{4 \times 3 \times 3 \times 10.5}{3} = 4 \times 3 \times 10.5$ $n = 12 \times 10.5 = 126$ ### Exam Strategy & Shortcut Recognize ratio multiples immediately. $10.5$ is exactly $3$ times $3.5$. The equation for $n$ simplifies to: $n = 4 \times (\frac{R}{r})^2 \times (\frac{R}{h}) = 4 \times (3)^2 \times (\frac{10.5}{3}) = 36 \times 3.5 = 126$. ### Common Pitfall Converting decimal numbers to fractions (e.g., $21/2$ and $7/2$) works but can sometimes increase the chance of flipping numerator/denominator incorrectly. Rely on direct division if the numbers form clean integer ratios like $10.5 / 3.5 = 3$. ### Final Answer Therefore, the correct answer is **126**.