The correct answer is **D. $108$** **Step-by-step derivation:** 1. **Identify the given information and relevant formulas:** * Radius of the sphere $= R$. * Volume of a sphere ($V_S$) $= \frac{4}{3} \pi R^3$. * Base radius of each cone $= r$. * Height of each cone ($h_c$) $= r$ (since height is equal to its base radius). * Volume of a cone ($V_C$) $= \frac{1}{3} \pi (\text{base radius})^2 (\text{height}) = \frac{1}{3} \pi r^2 h_c$. * Relationship between sphere and cone radii: $R = 3r$. * The sphere is melted and recast into $N$ cones, which means the total volume remains conserved. 2. **Calculate the volume of the sphere in terms of $r$:** Given $R = 3r$, substitute this into the sphere's volume formula: $V_S = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (3r)^3$ $V_S = \frac{4}{3} \pi (27r^3)$ $V_S = 36 \pi r^3$ 3. **Calculate the volume of one cone in terms of $r$:** Given $h_c = r$, substitute this into the cone's volume formula: $V_C = \frac{1}{3} \pi r^2 h_c = \frac{1}{3} \pi r^2 (r)$ $V_C = \frac{1}{3} \pi r^3$ 4. **Equate the total volume of cones to the volume of the sphere:** Since the sphere is melted and recast into $N$ identical cones, the total volume of the cones must be equal to the volume of the sphere. $N \times V_C = V_S$ $N \times \left(\frac{1}{3} \pi r^3\right) = 36 \pi r^3$ 5. **Solve for $N$:** Divide both sides by $\pi r^3$: $N \times \frac{1}{3} = 36$ $N = 36 \times 3$ $N = 108$ Therefore, $108$ cones can be formed. **Analysis of incorrect options:** * **A. $27$**: This would be the answer if one incorrectly assumed both volumes were simply proportional to the cube of their radii without the $\frac{4}{3}$ and $\frac{1}{3}$ factors, i.e., $R^3 / r^3 = (3r)^3 / r^3 = 27$. Or if one mistakenly used $V_S = \pi R^3$ and $V_C = \pi r^3$. * **B. $54$**: This could be obtained if one incorrectly assumed the sphere was a hemisphere (volume $= \frac{2}{3} \pi R^3$) and then divided by the cone's volume (correctly or incorrectly). Specifically, if $V_S = \frac{2}{3} \pi R^3$ and $V_C = \frac{1}{3} \pi r^3$, then $N = \frac{\frac{2}{3} \pi (3r)^3}{\frac{1}{3} \pi r^3} = \frac{2 \times 27}{1} = 54$. * **C. $81$**: This would be the answer if one used $V_S = \pi R^3$ (missing the $\frac{4}{3}$ factor) but correctly used $V_C = \frac{1}{3} \pi r^3$. Then $N = \frac{\pi (3r)^3}{\frac{1}{3} \pi r^3} = \frac{27}{\frac{1}{3}} = 27 \times 3 = 81$.