# Step-by-Step Solution ## **Restating the Problem** You need to choose between two metallic powders (same material) for powder metallurgy: - **Option 1:** Cylindrical particles, height (L) = diameter (D) - **Option 2:** Cubic particles, side dimension = L **Which one do you choose, and why?** --- ## **Step 1: Understanding Powder Metallurgy Needs** In powder metallurgy, **higher surface area per unit volume** is generally preferred because: - It improves **sintering** (better bonding at lower temperatures). - Increases **reactivity**. So, **choose the particle with the highest surface area to volume ratio**. --- ## **Step 2: Calculate Surface Area to Volume Ratios** ### **A. Cylinder (height = diameter = L)** - **Volume of cylinder:** \[ V_{\text{cyl}} = \pi r^2 h \] Where \( h = L \), \( D = L \), so \( r = \frac{L}{2} \). \[ V_{\text{cyl}} = \pi \left(\frac{L}{2}\right)^2 L = \pi \frac{L^2}{4} L = \frac{\pi}{4} L^3 \] - **Surface Area of cylinder:** \[ A_{\text{cyl}} = 2\pi r h + 2\pi r^2 \] Substitute \( r = \frac{L}{2}, h = L \): \[ A_{\text{cyl}} = 2\pi \frac{L}{2} L + 2\pi \left(\frac{L}{2}\right)^2 = \pi L^2 + \frac{\pi}{2} L^2 = \frac{3\pi}{2} L^2 \] - **Surface Area to Volume Ratio:** \[ \frac{A_{\text{cyl}}}{V_{\text{cyl}}} = \frac{\frac{3\pi}{2}L^2}{\frac{\pi}{4}L^3} = \frac{3\pi}{2} \cdot \frac{4}{\pi} \cdot \frac{1}{L} = 6 \cdot \frac{1}{L} \] --- ### **B. Cube (side = L)** - **Volume of cube:** \[ V_{\text{cube}} = L^3 \] - **Surface Area of cube:** \[ A_{\text{cube}} = 6L^2 \] - **Surface Area to Volume Ratio:** \[ \frac{A_{\text{cube}}}{V_{\text{cube}}} = \frac{6L^2}{L^3} = \frac{6}{L} \] --- ## **Step 3: Comparison and Final Decision** - **Cylinder:** \(\frac{6}{L}\) - **Cube:** \(\frac{6}{L}\) **Both have the same surface area to volume ratio!** --- ## **Step 4: Any Other Considerations?** Since the ratio is the same, either shape is **equally good** from a surface area to volume perspective. In practice, other factors (like flowability, packing density, manufacturing cost) might matter, but **based solely on surface area to volume ratio, either is acceptable**. --- ## **Final Answer** > **You can choose either the cylindrical or the cubic particles, as both have the same surface area to volume ratio (\(\frac{6}{L}\)).** > **Both are equally suitable for powder metallurgy applications based on this criterion.**