12 spheres of the same size are made from melting a solid cylinder of 16 cm diameter and 2 cm height. The diameter of each sphere is

### Concept & Fractional Volume When melting a large solid into multiple identical smaller ones, the total volume is perfectly distributed. $$Volume_{cylinder} = n \times Volume_{sphere}$$ ### Step-by-Step Solution 1. Identify the dimensions of the initial cylinder: Diameter = $16 \text{ cm} \implies$ radius $r = 8 \text{ cm}$. Height $h = 2 \text{ cm}$. 2. Calculate the total volume of the cylinder: $V = \pi r^2 h = \pi (8)^2 (2) = \pi \times 64 \times 2 = 128\pi \text{ cm}^3$. 3. This volume is divided equally into 12 spheres. 4. Let the radius of one sphere be $R$. The volume of 12 spheres is $12 \times (\frac{4}{3}\pi R^3) = 16\pi R^3$. 5. Equate the total volume: $16\pi R^3 = 128\pi$. 6. Cancel $\pi$ and divide by 16: $R^3 = \frac{128}{16} = 8$. 7. Solve for the radius: $R = \sqrt[3]{8} = 2 \text{ cm}$. 8. The question asks for the diameter: $Diameter = 2 \times R = 2 \times 2 = 4 \text{ cm}$. ### Exam Strategy & Shortcut Set up the single equation $12 \times \frac{4}{3}\pi R^3 = \pi (8)^2(2)$. Immediately cancel $\pi$. $16 R^3 = 64 \times 2 = 128$. $R^3 = 8 \implies R = 2$. Always highlight whether the question asks for radius or diameter to avoid missing the final step. ### Common Pitfall Solving the equation perfectly to find the radius ($2 \text{ cm}$) and immediately selecting option (b), completely forgetting that the question explicitly asked for the **diameter**. ### Final Answer Therefore, the correct answer is **4 cm**.