Correct Answer: C **Step 1: Write down the formula for the volume of the sphere.** The volume of a sphere with radius $R$ is given by: $V_{\text{sphere}} = \frac{4}{3}\pi R^3$ **Step 2: Write down the formula for the volume of a cone.** The volume of a cone with radius $r$ and height $h$ is given by: $V_{\text{cone}} = \frac{1}{3}\pi r^2 h$ **Step 3: Establish the given relationships between the dimensions.** According to the problem statement, we have two key relationships: 1. The radius of the sphere is three times the radius of each cone: $R = 3r$ 2. The height of each cone is four times its radius: $h = 4r$ **Step 4: Substitute the relationships into the volume formulas to express them in terms of a common variable ($r$).** Substitute $R = 3r$ into the sphere's volume formula: $V_{\text{sphere}} = \frac{4}{3}\pi (3r)^3$ $V_{\text{sphere}} = \frac{4}{3}\pi (27r^3)$ $V_{\text{sphere}} = 36\pi r^3$ Substitute $h = 4r$ into the cone's volume formula: $V_{\text{cone}} = \frac{1}{3}\pi r^2 (4r)$ $V_{\text{cone}} = \frac{4}{3}\pi r^3$ **Step 5: Equate the total volume of the cones to the volume of the sphere.** When a solid metallic object is melted and recast into other solids, the total volume of the material remains constant. Therefore, the volume of the original sphere is equal to the sum of the volumes of $N$ identical cones: $V_{\text{sphere}} = N \times V_{\text{cone}}$ Substitute the expressions for $V_{\text{sphere}}$ and $V_{\text{cone}}$ derived in Step 4: $36\pi r^3 = N \times \left(\frac{4}{3}\pi r^3\right)$ **Step 6: Solve for N.** To find $N$, we can divide both sides of the equation by $\frac{4}{3}\pi r^3$. We can do this because $r$ must be non-zero for a physical cone: $N = \frac{36\pi r^3}{\frac{4}{3}\pi r^3}$ Cancel out the common term $\pi r^3$ from the numerator and denominator: $N = \frac{36}{\frac{4}{3}}$ To divide by a fraction, multiply by its reciprocal: $N = 36 \times \frac{3}{4}$ Perform the multiplication: $N = 9 \times 3$ $N = 27$ Thus, $27$ identical solid cones can be made from the melted sphere. The final answer is $\boxed{\text{27}}$.