A metallic cone of radius 12 cm and height 24 cm is melted and made into spheres of radius 2 cm each. How many spheres are there?

### Concept & Volume Equivalence in Recasting When a large shape is melted down to create multiple smaller shapes, the total volume of the original shape is divided by the volume of one smaller shape to determine the total quantity produced. $$n = \frac{V_{cone}}{V_{sphere}}$$ ### Step-by-Step Solution 1. **Volume of the Original Cone:** $r = 12\text{ cm}, h = 24\text{ cm}$ $V_{cone} = \frac{1}{3}\pi r^2h = \frac{1}{3}\pi (12)^2(24) = 8\pi \times 144 = 1152\pi\text{ cm}^3$ 2. **Volume of One Small Sphere:** $R = 2\text{ cm}$ $V_{sphere} = \frac{4}{3}\pi R^3 = \frac{4}{3}\pi (2)^3 = \frac{32}{3}\pi\text{ cm}^3$ 3. **Calculate Number of Spheres ($n$):** $n = \frac{1152\pi}{\frac{32}{3}\pi} = 1152 \times \frac{3}{32}$ $n = 36 \times 3 = 108$ ### Exam Strategy & Shortcut Set up the fraction before fully expanding the multiplications: $n = \frac{\frac{1}{3}\pi(12 \times 12 \times 24)}{\frac{4}{3}\pi(2 \times 2 \times 2)}$ Cancel the $\frac{\pi}{3}$: $n = \frac{12 \times 12 \times 24}{4 \times 8} = \frac{12 \times 12 \times 24}{32}$ Simplify: $n = \frac{144 \times 24}{32} = \frac{144 \times 3}{4} = 36 \times 3 = 108$. ### Common Pitfall A standard calculation error occurs when dividing by a fraction; always remember to multiply by the reciprocal (e.g., multiply by $3/32$ instead of dividing by $32/3$). ### Final Answer Therefore, the correct answer is **108**.