Question 1178894
Let's solve this problem step-by-step:

**a) Diameter of the Standard Cylinder:**

1.  **Volume Formula:** The volume of a cylinder is V = πr²h, where V is volume, r is radius, and h is height (length).
2.  **Given Values:**
    * V = 0.015 m³
    * h = 75 cm = 0.75 m (convert to meters for consistent units)
3.  **Solve for Radius (r):**
    * 0.015 = πr²(0.75)
    * r² = 0.015 / (0.75π)
    * r² ≈ 0.0063661977
    * r ≈ √0.0063661977 ≈ 0.079788 m
4.  **Convert Radius to Centimeters:**
    * r ≈ 0.079788 m * 100 cm/m ≈ 7.9788 cm
5.  **Calculate Diameter (d):**
    * d = 2r ≈ 2 * 7.9788 cm ≈ 15.9576 cm
6.  **Round to Nearest Centimeter:**
    * d ≈ 16 cm

**Therefore, the diameter of the standard cylinder is approximately 16 cm.**

**b) Dimensions of the New Cylinder (Less Material):**

To minimize material, we want to minimize the surface area for a given volume. This occurs when the height of the cylinder is equal to the diameter (h = 2r).

1.  **Volume Formula:** V = πr²h
2.  **Given Values:**
    * V = 0.015 m³
    * h = 2r
3.  **Substitute h = 2r:**
    * 0.015 = πr²(2r)
    * 0.015 = 2πr³
    * r³ = 0.015 / (2π)
    * r³ ≈ 0.002387324
    * r ≈ ³√0.002387324 ≈ 0.13365 m
4.  **Convert Radius to Centimeters:**
    * r ≈ 0.13365 m * 100 cm/m ≈ 13.365 cm
5.  **Calculate Height (h):**
    * h = 2r ≈ 2 * 13.365 cm ≈ 26.73 cm
6.  **Round to Nearest Tenth of a Centimeter:**
    * r ≈ 13.4 cm
    * h ≈ 26.7 cm
7.  **Calculate Surface Area (SA):**
    * SA = 2πr² + 2πrh
    * SA = 2π(13.365)² + 2π(13.365)(26.73)
    * SA ≈ 2π(178.623225) + 2π(357.24645)
    * SA ≈ 1122.38 + 2244.75
    * SA ≈ 3367.13 cm²
8.  **Round to Nearest Square Centimeter:**
    * SA ≈ 3367 cm²

**Therefore, the new cylinder dimensions are approximately 13.4 cm radius and 26.7 cm height. The surface area is approximately 3367 cm².**

**c) Packing Arrangement (Least Volume and Surface Area):**

1.  **Standard Cylinder Dimensions:**
    * Diameter (d) = 16 cm
    * Radius (r) = 8 cm
    * Height (h) = 75 cm
2.  **Box A (Identical Rows):**
    * Dimensions:
        * Length: 5 * d = 5 * 16 cm = 80 cm
        * Width: 4 * d = 4 * 16 cm = 64 cm
        * Height: 75 cm
    * Volume: 80 cm * 64 cm * 75 cm = 384,000 cm³
3.  **Box B (Staggered Rows):**
    * Dimensions:
        * Length: 5 * d = 80 cm
        * Width: d + 3 * (d * √3 / 2) = 16 + 3 * (16 * √3 / 2) ≈ 16 + 41.57 = 57.57 cm
        * Height: 75 cm
    * Volume: 80 cm * 57.57 cm * 75 cm = 345,420 cm³

**Comparison:**

* **Volume:** Box B (staggered) has a smaller volume (345,420 cm³) compared to Box A (384,000 cm³).
* **Surface Area:** Box B will also have a smaller surface area, as it requires a smaller box overall.

**Conclusion:**

The **staggered packing arrangement (Box B)** will require the box of least volume and surface area.