Some solid metallic right circular cones, each with radius of the base 3 cm and height 4 cm, are melted to form a solid sphere of radius 6 cm. The number of right circular cones is

### Concept & Volume Division To find the number of smaller objects melted to form a larger object, divide the total volume of the larger object by the volume of a single smaller object. $$n = \frac{V_{sphere}}{V_{cone}}$$ ### Step-by-Step Solution 1. **Calculate Volume of One Cone:** $r = 3\text{ cm}, h = 4\text{ cm}$ $V_{cone} = \frac{1}{3}\pi r^2h = \frac{1}{3}\pi (3)^2(4) = 12\pi\text{ cm}^3$ 2. **Calculate Volume of Sphere:** $R = 6\text{ cm}$ $V_{sphere} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (6)^3 = \frac{4}{3}\pi (216) = 288\pi\text{ cm}^3$ 3. **Determine Number of Cones ($n$):** $n = \frac{288\pi}{12\pi} = 24$ ### Exam Strategy & Shortcut Set up the full equation before simplifying to cancel out terms early: $n = \frac{\frac{4}{3}\pi(6 \times 6 \times 6)}{\frac{1}{3}\pi(3 \times 3 \times 4)}$. The $\frac{\pi}{3}$ cancels out. $n = \frac{4 \times 6 \times 6 \times 6}{3 \times 3 \times 4} = \frac{6 \times 6 \times 6}{3 \times 3} = 2 \times 2 \times 6 = 24$. This is much faster and reduces arithmetic mistakes. ### Common Pitfall A common mistake is incorrectly calculating the initial volumes by ignoring the fractional constants ($\frac{4}{3}$ and $\frac{1}{3}$), assuming they will inherently cancel out the same way. ### Final Answer Therefore, the correct answer is **24**.